Polynomials

Students will:

(1.) Discuss polynomials.
(2.) Simplify polynomials.
(3.) Determine the domain of polynomial functions.
(4.) Determine the range of polynomial functions.
(5.) Evaluate polynomial functions for a given value.
(6.) Add polynomials.
(7.) Subtract polynomials.
(8.) Multiply polynomials.
(9.) Divide polynomials.
(10.) Discuss the Division Algorithm.
(11.) Check the solution of their division using the Division Algorithm.
(12.) Discuss the Remainder Theorem.
(13.) Discuss the Factor Theorem.
(14.) Factor polynomials.
(15.) Graph polynomial functions.
(16.) Determine the zeros of polynomial functions.
(17.) Determine the multiplicity of the zeros of polynomial functions.
(18.) Determine algebraically whether the graph of a polynomial function crosses or touches the x-axis.
(19.) Determine the intercepts of polynomial functions.
(20.) Determine the absolute extrema of polynomial functions.
(21.) Determine the relative extrema of polynomial functions.
(22.) Discuss the Rational Root Theorem.
(23.) Determine the end-behavior of the graphs of polynomial functions.
(24.) Analyze the graphs of polynomial functions.
(25.) Discuss the Descartes' Rule of Signs.
(26.) Solve polynomial problems using the TI-calculators series.
(27.) Solve polynomial problems using calculators.

Skills Measured/Acquired

(1.) Use of prior knowledge
(2.) Critical Thinking
(3.) Interdisciplinary connections/applications
(4.) Technology
(5.) Active participation through direct questioning
(6.) Research

Please note:
added to
subtracted from
multiplied by
divided by

Check for prior knowledge. Ask students about these terms.

Bring it to English: vary, constant, express, expression, equate, equal, equation, equality, equanimity, equity, addendum
Bring it to Physics: Prefixes: mono or uni = $1$, di or bi = $2$, tri = $3$, tetra or quad = $4$, penta = $5$, hexa = $6$, hepta = $7$, octa = $8$, nona = $9$, deca = $10$, hendeca = $11$, dodeca = $12$, etc.

Ask students to give examples of real-world scenarios where they have used any of the prefixes.
These include: unity, bilateral, triangle, tetrahedral, quadrilaterals (you can ask students to list examples of quadrilaterals - Geometry!), etc.

Bring it to Math: arithmetic, arithmetic operators, algebra, polynomial functions, polynomials, monomial, binomial, trinomial, quadrinomial, tetranomial, pentanomial, quintinomial, hexanomial, heptanomial, octanomial, nonanomial, decanomial, sum, difference, product, quotient, type of polynomial, degree of polynomial, Rational root theorem, root, solution, zeros, Descartes' Rule of signs, functions, linear functions, quadratic functions, cubic functions, slope, intercepts, y-intercept, x-intercept, factoring techniques, arithmetic operations, slope-intercept form, standard form, constant form, point-slope form, general form, vertex, axis, line of symmetry, symmetry, long division, synthetic division, Remainder theorem, Factor theorem, add, subtract, multiply, divide, augend, addend, minuend, subtrahend, multiplicand, multiplier, dividend, divisor, sum, difference, product, quotient, division algorithm, cubic, quartic, exponent, index, power, degree, order, quintic, pentic, hexic, sextic, heptic, septic, octic, nonic, decic, extrema, maxima, minima, multiplicity of zeros, relative extrema, absolute extrema, relative maximum, relative minimum, absolute maximum, absolute minimum, global extrema, local extrema, global minimum, global maximum, local minimum, local maximum, domain, range, FOIL (First Outer Inner Last), box method, etc.

Generally, "linear" implies that the exponent of the variable is 1
"quadratic" implies that the exponent of the variable is 2
"cubic" implies that the exponent of the variable is 3
"quartic" implies that the exponent of the variable is 4

Some students may ask the reason for the discrepancy between "quadratic" and "quartic".
Please explain.

The basic arithmetic operators are the addition symbol, $+$, the subtraction symbol, $-$, the multiplication symbol, $*$, and the division symbol, $\div$

Augend is the term that is being added to. It is the first term.

Addend is the term that is added. It is the second term.

Sum is the result of the addition.

$ For: \\[3ex] 3 + 7 = 10 \\[3ex] 3 = augend \\[3ex] 7 = addend \\[3ex] 10 = sum \\[3ex] $ Minuend is the term that is being subtracted from. It is the first term.

Subtrahend is the term that is subtracted. It is the second term.

Difference is the result of the subtraction.

$ For: \\[3ex] 3 - 7 = -4 \\[3ex] 3 = minuend \\[3ex] 7 = subtrahend \\[3ex] -4 = difference \\[3ex] $ Multiplier is the term that is multiplied by. It is the first term.

Multiplicand is the term that is multiplied. It is the second term.

Product is the result of the multiplication.

$ For: \\[3ex] 3 * 10 = 30 \\[3ex] 3 = multiplier \\[3ex] 10 = multiplicand \\[3ex] 30 = product \\[3ex] $ Dividend is the term that is being divided. It is the numerator.

Divisor is the term that is dividing. It is the denominator.

Quotient is the result of the division.

Remainder is the term remaining after the division.

$ For: \\[3ex] 12 \div 7 \\[3ex] = 1\dfrac{5}{7} \\[5ex] = 1 \:R\: 5 \\[3ex] 12 = dividend \\[3ex] 7 = divisor \\[3ex] 1 = quotient \\[3ex] 5 = remainder \\[3ex] $ A constant is something that does not change. In mathematics, numbers are usually the constants.

A variable is something that varies (changes). In Mathematics, alphabets are usually the variables.

A function is a relation in which each input value has a unique output value.
The unique output value means that an input value cannot have two or more output values.
However, two or more input values can have the same output value.

A relation is a set of ordered pairs in which there each input value has at least one output value.

A linear function is a function in which the highest exponent of the independent variable in the function is $1$

A quadratic function is an function in which the highest exponent of the independent variable in the function is $2$

A cubic function is an function in which the highest exponent of the independent variable in the function is $3$

A quartic function is an function in which the highest exponent of the independent variable in the function is $4$

A polynomial is a function of variable(s) with only non-negative integer exponents.

We can also define it as:
A polynomial is a function:
that is a combination of constants and/or variables,
and in which the variable(s) do not have negative exponents or fractional exponents.
Simply put, the exponents are only non-negative integer exponents.
The coefficient(s) are constants but the variable(s) are non-negative integers only.

For a function to be a polynomial function,
the constants can have negative exponents or fractional exponents.
However, the variable(s) cannot have negative exponents or fractional exponents.
Students should give examples of each case to demonstrate understanding.

A polynomial is in standard form if it is written in descending order of exponents of the variable.

The degree of a polynomial is defined as the:
highest exponent of the variable (if the polynomial has only one variable) OR
the greater of: the highest exponent of the variable and the sum of the exponents of the variables (if the polynomial has several variables)

A polynomial of degree 1 is known as a linear polynomial.
A polynomial of degree 2 is known as a quadratic polynomial.
A polynomial of degree 3 is known as a cubic polynomial.
A polynomial of degree 4 is known as a quartic polynomial.
A polynomial of degree 5 is known as a quintic polynomial.
A polynomial of degree 6 is known as a hexic or sextic polynomial.

The type of a polynomial is defined as the number of terms in the polynomial.
A polynomial that has one term is known as a monomial.
A polynomial that has two terms is known as a binomial.
A polynomial that has three terms is known as a trinomial.
A polynomial that has four terms is known as a quadrinomial or tetranomial.
A polynomial that has five terms is known as a quintinomial or pentanomial.
A polynomial that has six terms is known as a hexanomial.

Please note these terms:
added to
subtracted from
multiplied by
divided by

A polynomial is a function of variable(s) with only non-negative integer exponents.
In other words, a polynomial has:
(1.) At least one variable.
(2.) The variable has at least one exponent.
(3.) The exponent must be a non-negative integer.
The standard form of a polynomial is: $f(x) = a_nx^n + a_{n - 1}x^{n - 1} + a_{n - 2}x^{n - 2} + ... + a_1x + a_0$
where:
$f(x)$ is the dependent variable (output)
$x$ is the independent variable (input)
$n$, $n - 1$, $n - 2$, are the exponents
$a_nx^n$ is the first term
$a_{n - 1}x^{n - 1}$ is the second term
$a_{n - 2}x^{n - 2}$ is the third term
$a_0$ is the last term
$a_n$, $a_{n - 1}$, $a_{n - 2}$, $a_1$ are the coefficients of the x-variable

A power function is a function of the form: $f(x) = a_n x^n$
where:
$f(x)$ is the dependent variable (output)
$x$ is the independent variable (input)
$n$ is the exponent
$a_n x_n$ is the term in the function
$a_n$ is the coefficient of the x-variable
$a$ and $n$ are real numbers.

However, for a polynomial, $n$ must be a non-negative integer. It cannot be any real number.
So, the main difference between a polynomial and a power function is a polynomial must have only non-negative integer exponents while a power function can have any real-number exponent.
In this regard, we can also define a polynomial as the sum of several power functions having only non-negative integer exponents.

Why Polynomials? How Did Polynomials Come Up?
Student: I know that an integer is a whole number including 0
But, what is a non-negative integer?
Teacher: Good question.
Is zero a positive or negative integer?
Student: Neither.
0 is used to separate the positive numbers from the negative numbers.
Teacher: Correct.
What if I wanted to include 0 with the positive integers?
What term do you think I should call it?
Student: I guess ...non-negative?
Teacher: That is right.
Non-negative integers are: zero and the positive integers: 0, 1, 2, 3, 4, ..., ∞
Student: So, the difference between a positive integer and a non-negative integer is that
A positive integer does not include 0 but a non-negative integer includes 0
Teacher: That is correct.
I would say it this way: that the non-negative integer includes 0 and the positive integer while the positive integer does not include 0.
I would emphasize that non-negative integers includes zero and positive integers.
In that same sense...
The difference between negative integers and non-positive integers is:
Student: Negative integers contain only negative numbers (without zero) but Non-positive integers includes zero and the negative integers.
Teacher: Correct.
Student: How did Polynomials come up?
Teacher: Mathematicians wanted to study functions that have variables with exponents that are non-negative integers only.
They wanted to study the characteristics of those kind of functions...functions with exponents that are either zero or positive...no negative exponents (inverse functions), no fractional exponents (radical functions), no exponents with variables (such as exponential functions), no other kinds of functions such as trigonometric functions, logarithmic functions, absolute value functions, etc.
Just functions with simple integer exponents...
That leads to the study of Polynomials
Those kind of functions (functions of variables with non-negative integer exponents) are Polynomial Functions or just Polynomials.
Let us begin. 😊

A polynomial is said to be in standard form if it is written in descending order of exponents of the variable.
When a polynomial is written in standard form, the leading term of the polynomial is usually the first nonzero term with the variable. It is the term with the highest exponent of the variable.
The constant term is the term without the variable.

Example 1:
One of these polynomials is written in standard form.
Which one?

$ A.\;\; f(p) = 3 - 2p^2 + 5p \\[3ex] B.\;\; f(p) = 5p - 2p^2 + 3 \\[3ex] C.\;\; f(p) = -2p^2 + 5p + 3 \\[3ex] D.\;\; f(p) = 5p + 3 - 2p^2 \\[3ex] E.\;\; f(p) = -2p^2 + 3 + 5p \\[3ex] F.\;\; f(p) = 3 + 5p - 2p^2 $

A polynomial can have one or more variables.
If the polynomial has only one variable, the degree of the polynomial is the highest exponent of the variable.
If the polynomial has more than one variable, the degree of the polynomial is the greater of: the highest exponent of the variable and the sum of the exponents of the variables.

The type of a polynomial is the number of terms in the polynomial.

After reviewing these terms in the Definition and Introduction sections of the topic, it is important to check your understanding.

Example 2:
(1.) Please indicate whether each function in the table below is a polynomial or not.
If it is not a polynomial, please state so and move to the next question where applicable.
If it is a polynomial, please state:
(2.) the number of variable(s) and the variable(s) in the polynomial.
(3.) whether it is written in standard form. If it is not in standard form, write it in standard form.
(4.) the type and degree of the polynomial.
(5.) the leading term.
(6.) the leading coefficient.
(7.) the constant term as applicable.

Hint:
(1.) Does the function/expression have a variable?
OR
Can the function/expression be expressed as a variable?
Keep in mind that any constant can be expressed as a variable by including the variable with it and raising that variable to exponent zero...Law 3...Exp

(2.) Does the variable have any exponent?
Keep in mind that any variable has an exponent. If you do not see the exponent, it means the exponent is 1 ...Law 4...Exp

(3.) Is the exponent a non-negative integer?
If any of the exponents is not a non-negative integer, then the function is not a polynomial.

Given:
(1.) A polynomial with a variable
(2.) A value (constant or variable)
To Evaluate: the polynomial at that value
We substitute the variable in the polynomial with the given value.
This means that whenever you see the variable in the polynomial, replace it with the given value and simplify.
Please note that the given value can be a constant or a variable.

Example 1:

Work students through the addition and subtraction of variables.
Use real-world examples.
Let them know that the reasoning/thought process discussed below applies to addition and subtraction.
It's a different reasoning when we discuss multiplication.

Recall:
2 cups + 3 cups = 5 cups
Let cup = c
This implies that:
2c + 3c = 5c

What about 2 cups + 3 pens?
2 cups + 3 pens = 2 cups + 3 pens because a cup is different from a pen.
Technically, we can say that: 2 cups + 3 pens = 5 items
However, that is not our goal.
Our goal is to add or subtract same thing(s) to give us same thing(s).
We cannot add or subtract different things.
So:
2c + 3p = 2c + 3p

What about 2 cups + 3?

Teacher: Samuel, can you give me 3?
Samuel: 3 what?
Teacher: Notice how you said 3 what? what? what?
Notice the importance of including units to physical quantities.
Did you notice that 3 could mean several things?
It could mean $3 or 3 cups or 3 pens or 3 computers or 3 colleges or 3 universities, etc. Hence, we cannot add 3 to 2 cups.
If it was 3 cups, then we can add it to 2 cups to give 5 cups.
However, because we are not sure of the 3, we cannot add it to 2 cups.
Therefore, 2 cups + 3 still gives us 2 cups + 3
Use this moment to teach students the importance of writing units for measurements.

To add polynomials:
(1.) Apply the Distributive Property to each polynomial as applicable.

(2.) Ensure that the polynomials are in standard form
If any of them is not in standard form, write it in standard form.

(3.) If you use the Vertical Method to add the polynomials, complete any missing term and include zero as the coefficient of that missing term.

(4.) Count the number of terms to make sure you did not miss/skip any term.

(5.) Add the terms with same exponents
In other words, add like terms

(6.) Write the sum and any remaining terms.

(7.) Put the final answer in standard form.

We can use the Horizontal Method or the Vertical Method to add polynomials.
Use any method you prefer.

Example 1: Simplify: $(7x^3 - x^2 - 3x - 7) + (6x^3 - 2x^2 + 4x + 3)$
Solution 1:
Both polynomials are in standard form.

$ \underline{Horizontal\;\;Method} \\[3ex] (7x^3 - x^2 - 3x - 7) + (6x^3 - 2x^2 + 4x + 3) \\[3ex] = 1(7x^3 - x^2 - 3x - 7) + 1(6x^3 - 2x^2 + 4x + 3) \\[3ex] = 7x^3 - x^2 - 3x - 7 + 6x^3 - 2x^2 + 4x + 3 \\[3ex] = 7x^3 + 6x^3 - x^2 - 2x^2 - 3x + 4x - 7 + 3 \\[3ex] = 13x^3 - 3x^2 + x - 4 \\[5ex] \underline{Vertical\;\;Method} \\[3ex] \begin{array}{c} 7x^3 - x^2 - 3x - 7 \\ +\hspace{9em} \\ 6x^3 - 2x^2 + 4x + 3 \\ \hline 13x^3 - 3x^2 + x - 4 \\ \hline \end{array} $

Example 2: Add the polynomials: $(x^2 - 6x^5 - 12 + 3x)$ and $(6x^3 + x^2 - 8 + 3x^4)$
Solution 2:
Both polynomials are not in standard form.
Let us put each of them in standard form.
Add the polynomials: $(-6x^5 + x^2 + 3x - 12) + (3x^4 + 6x^3 + x^2 - 8)$

$ \underline{Horizontal\;\;Method} \\[3ex] (-6x^5 + x^2 + 3x - 12) + (3x^4 + 6x^3 + x^2 - 8) \\[3ex] = 1(-6x^5 + x^2 + 3x - 12) + 1(3x^4 + 6x^3 + x^2 - 8) \\[3ex] = -6x^5 + x^2 + 3x - 12 + 3x^4 + 6x^3 + x^2 - 8 \\[3ex] = -6x^5 + 3x^4 + 6x^3 + x^2 + x^2 + 3x - 12 - 8 \\[3ex] = -6x^5 + 3x^4 + 6x^3 + 2x^2 + 3x - 20 \\[5ex] $ To use the Vertical Method, it is highly recommended we complete both polynomials.
This means that we have to include any missing term with 0 as the coefficient.

$ \underline{Vertical\;\;Method} \\[3ex] \begin{array}{c} -6x^5 + 0x^4 + 0x^3 + x^2 + 3x - 12 \\ +\hspace{16em} \\ 0x^5 + 3x^4 + 6x^3 + x^2 + 0x - 8 \\ \hline -6x^5 + 3x^4 + 6x^3 + 2x^2 + 3x - 20 \\ \hline \end{array} $

Example 3: Determine the sum of the polynomials: $-3(7p - p^3 + 12p^2)$ and $-5(-8 - 9p^2 + 2p^4 - p)$
Solution 3:
First Step: The Distributive Property is applicable to each of them.
So, let us distribute −3 to the first polynomial and distribute −5 to the second polynomial.
This gives us:
$-21p + 3p^3 - 36p^2$ and $40 + 45p^2 - 10p^4 + 5p$
Both polynomials are not in standard form.
Second Step: Let us put each of them in standard form.
Add the polynomials: $3p^3 - 36p^2 - 21p$ and $-10p^4 + 45p^2 + 5p + 40$

$ \underline{Horizontal\;\;Method} \\[3ex] (3p^3 - 36p^2 - 21p) + (-10p^4 + 45p^2 + 5p + 40) \\[3ex] 1(3p^3 - 36p^2 - 21p) + 1(-10p^4 + 45p^2 + 5p + 40) \\[3ex] = 3p^3 - 36p^2 - 21p - 10p^4 + 45p^2 + 5p + 40 \\[3ex] = -10p^4 + 3p^3 - 36p^2 + 45p^2 - 21p + 5p + 40 \\[3ex] = -10p^4 + 3p^3 + 9p^2 - 16p + 40 \\[5ex] $ To use the Vertical Method, it is highly recommended we complete both polynomials.
This means that we have to include any missing term with 0 as the coefficient.

$ \underline{Vertical\;\;Method} \\[3ex] \begin{array}{c} 0p^4 + 3p^3 - 36p^2 - 21p + 0 \\ +\hspace{16em} \\ -10p^4 + 0p^3 + 45p^2 + 5p + 40 \\ \hline -10p^4 + 3p^3 + 9p^2 - 16p + 40 \\ \hline \end{array} $

Subtraction of Polynomials
The same principles we used in the addition of polynomials also apply to the subtraction of polynomials.

To subtract polynomials:
(1.) Apply the Distributive Property to each polynomial as applicable.

(2.) Ensure that the polynomials are in standard form
If any of them is not in standard form, write it in standard form.

(3.) If you use the Vertical Method to subtract the polynomials, complete any missing term and include zero as the coefficient of that missing term.

(4.) Count the number of terms to make sure you did not miss/skip any term.

(5.) Subtract the terms with same exponents
In other words, subtract like terms

(6.) Write the difference and any remaining terms.

(7.) Put the final answer in standard form.

We can use the Horizontal Method or the Vertical Method to subtract polynomials.
Use any method you prefer.

Example 4: Evaluate: $(7x^3 - x^2 - 3x - 7) - (6x^3 - 2x^2 + 4x + 3)$
Solution 4:
Both polynomials are in standard form.

$ \underline{Horizontal\;\;Method} \\[3ex] (7x^3 - x^2 - 3x - 7) - (6x^3 - 2x^2 + 4x + 3) \\[3ex] = 1(7x^3 - x^2 - 3x - 7) - 1(6x^3 - 2x^2 + 4x + 3) \\[3ex] = 7x^3 - x^2 - 3x - 7 - 6x^3 + 2x^2 - 4x - 3 \\[3ex] = 7x^3 - 6x^3 - x^2 + 2x^2 - 3x - 4x - 7 - 3 \\[3ex] = x^3 + x^2 - 7x - 10 \\[5ex] \underline{Vertical\;\;Method} \\[3ex] \begin{array}{c} 7x^3 - x^2 - 3x - 7 \\ -\hspace{9em} \\ 6x^3 - 2x^2 + 4x + 3 \\ \hline x^3 + x^2 - 7x - 10 \\ \hline \end{array} $

Example 5: Find the difference between the polynomials: $(x^2 - 6x^5 - 12 + 3x)$ and $(6x^3 + x^2 - 8 + 3x^4)$
Solution 5:
Both polynomials are not in standard form.
Let us put each of them in standard form and subtract them, so we can find the difference.
$(-6x^5 + x^2 + 3x - 12) - (3x^4 + 6x^3 + x^2 - 8)$

$ \underline{Horizontal\;\;Method} \\[3ex] (-6x^5 + x^2 + 3x - 12) - (3x^4 + 6x^3 + x^2 - 8) \\[3ex] = 1(-6x^5 + x^2 + 3x - 12) - 1(3x^4 + 6x^3 + x^2 - 8) \\[3ex] = -6x^5 + x^2 + 3x - 12 - 3x^4 - 6x^3 - x^2 + 8 \\[3ex] = -6x^5 - 3x^4 - 6x^3 + x^2 - x^2 + 3x - 12 + 8 \\[3ex] = -6x^5 - 3x^4 - 6x^3 + 3x - 4 \\[5ex] $ To use the Vertical Method, it is highly recommended we complete both polynomials.
This means that we have to include any missing term with 0 as the coefficient.

$ \underline{Vertical\;\;Method} \\[3ex] \begin{array}{c} -6x^5 + 0x^4 + 0x^3 + x^2 + 3x - 12 \\ -\hspace{16em} \\ 0x^5 + 3x^4 + 6x^3 + x^2 + 0x - 8 \\ \hline -6x^5 - 3x^4 - 6x^3 + 0x^2 + 3x - 4 \\ \hline \end{array} \\[3ex] Answer = -6x^5 - 3x^4 - 6x^3 + 3x - 4 $

Example 6: Subtract $-5(-8 - 9p^2 + 2p^4 - p)$ from $-3(7p - p^3 + 12p^2)$

This implies: $-3(7p - p^3 + 12p^2) - -5(-8 - 9p^2 + 2p^4 - p)$
Solution 6:
We can use at least two methods (Horizontal Method and Vertical Method) to do this question in two different ways (First Approach and Second Approach).
Choose whatever method and whatever approach you prefer.

(a.) First Approach:
Use the fact that the two negatives make a positive,
distribute −3 to the first polynomial
distribute 5 to the second polynomial
add the two results to get the final answer.

(b.) Second Approach:
distribute −3 to the first polynomial and −5 to the second polynomials
subtract the two results to get the final answer.

$ \underline{Horizontal\;\;Method\;\;and\;\;First\;\;Approach} \\[3ex] -3(7p - p^3 + 12p^2) - -5(-8 - 9p^2 + 2p^4 - p) \\[3ex] = -3(7p - p^3 + 12p^2) + 5(-8 - 9p^2 + 2p^4 - p) \\[3ex] = -21p + 3p^3 - 36p^2 - 40 - 45p^2 + 10p^4 - 5p \\[3ex] = -21p + 3p^3 - 36p^2 - 40 - 45p^2 + 10p^4 - 5p \\[3ex] = 10p^4 + 3p^3 - 36p^2 - 45p^2 - 21p - 5p - 40 \\[3ex] = 10p^4 + 3p^3 - 81p^2 - 26p - 40 \\[5ex] \underline{Horizontal\;\;Method\;\;and\;\;Second\;\;Approach} \\[3ex] -3(7p - p^3 + 12p^2) - -5(-8 - 9p^2 + 2p^4 - p) \\[3ex] = -21p + 3p^3 - 36p^2 - (40 + 45p^2 - 10p^4 + 5p) \\[3ex] = -21p + 3p^3 - 36p^2 - 40 - 45p^2 + 10p^4 - 5p \\[3ex] = 10p^4 + 3p^3 - 36p^2 - 45p^2 - 21p - 5p - 40 \\[3ex] = 10p^4 + 3p^3 - 81p^2 - 26p - 40 \\[5ex] For\;\;the\;\;Vertical\;\;Method: \\[3ex] \underline{Distributive\;\;Property\;\;and\;\;Standard\;\;Form} \\[3ex] -3(7p - p^3 + 12p^2) \\[3ex] = -21p + 3p^3 - 36p^2 \\[3ex] = 3p^3 - 36p^2 - 21p \\[5ex] -5(-8 - 9p^2 + 2p^4 - p) \\[3ex] = 40 + 45p^2 - 10p^4 + 5p \\[3ex] = -10p^4 + 45p^2 + 5p + 40 \\[5ex] 5(-8 - 9p^2 + 2p^4 - p) \\[3ex] = -40 - 45p^2 + 10p^4 - 5p \\[3ex] = 10p^4 - 45p^2 - 5p - 40 \\[5ex] \underline{Vertical\;\;Method\;\;and\;\;First\;\;Approach} \\[3ex] \begin{array}{c} 0p^4 + 3p^3 - 36p^2 - 21p + 0 \\ +\hspace{16em} \\ 10p^4 + 0p^3 - 45p^2 - 5p - 40 \\ \hline 10p^4 + 3p^3 - 81p^2 - 26p - 40 \\ \hline \end{array} $

$ \underline{Vertical\;\;Method\;\;and\;\;Second\;\;Approach} \\[3ex] \begin{array}{c} 0p^4 + 3p^3 - 36p^2 - 21p + 0 \\ -\hspace{16em} \\ -10p^4 + 0p^3 + 45p^2 + 5p + 40 \\ \hline 10p^4 + 3p^3 - 81p^2 - 26p - 40 \\ \hline \end{array} $

Given: Two polynomials
To Find: The product
Multiplying Two Polynomials require that we multiply each term of one polynomial with each term of the other polynomial

Student: What if we were given three polynomials and asked to find the product?
Teacher: Good question.
In that case, do it two at a time.
Multiply the first two.
Find the product.
Then, multiply the third with that product.
The same principle apply to the multiplication of many polynomials.
It is better to multiply two at a time.
Find the product. Use it to multiply the third.
Find the product. Use it to multiply the fourth.
and so on and so forth.

There are at least four methods that we can use to multiply two polynomials.
They are:
(1.) Direct Multiplication (Horizontal Method).
This is used to multiply any two polynomials.
It is better to write each multiplication (multiplication of each term of the first polynomial with each term of the second polynomial) line-by-line.
In other words, do each multiplication on a new line
Then, combine all individual products to find the final product.
Some people prefer to write it horizontally on one line rather than multiple lines. But, I encourage you to write in multiple lines if you prefer to use this method. Be it as it may, do what works best for you.

(2.) FOIL (First — Outer — Inner — Last) Method.
This method is used to multiply only two binomials.
It is similar to the Horizontal Method but in the sense that it is used for only two binomials.
We multiply the:
(a.) First: first term in the first binomial with the first term in the second binomial
(b.) Outer: first term in the first binomial with the second term in the second binomial
(c.) Inner: second term in the first binomial with the first term in the second binomial
(d.) Last: second term in the first binomial with the second term in the second binomial

(3.) Box Method.
This method is used to multiply any two polynomials.
We set it up in the form of a box with compartments, putting one polynomial as the length of the box and the other polynomial as the width of the box.
We make sure to include the signs with the coefficients for each polynomial.
Each term of the polynomial takes up a space accordingly/respectively in the length and the width of the box.
The product of a term in the length and the corresponding term in the width is written in each compartment in the box.
Unless asked otherwise, it is highly recommended that both polynomials are set in standard form before using this method.

(4.) Vertical Method.
To use this method, each polynomial must be written in standard form.
Recall how we multiplied numbers in the elementary school. How did we set it up? Vertically right?
Then, we set up the two polynomials vertically just as we set up the multiplication of two numbers in the elementary school. 😊
We write the first polynomial on the first line, then write the multiplication symbol on the second line, and the second polynomial on the third line.
We begin the multiplication from the back...multiplying the last term of the second polynomial with the last term of the first polynomial.

Let us do some examples using each method.
Use any method you prefer.

Example 1:

Identifying Graphs of Polynomials
(1.) The graph of every polynomial function is both smooth and continuous.
(2.) The graph is smooth implies that it contains no sharp corners or cusps.
(3.) The graph is continuous implies that it has no gaps or holes and can be drawn without lifting pencil from paper.

Exercise 1:
Identify whether these graphs are polynomials or not.
Give reasons.

Answer: (Click to Show/Hide)

Graph A is a polynomial function because the graph is smooth and continuous. It is a parabola.
Graph B is not a polynomial because it is not smooth: there is a sharp corner between the curved part and the straight part.
Graph C is not a polynomial because it is not smooth: there is a sharp corner between the negative slope and the positive slope sections of the graph. It is the graph of an absolute value function.
Graph D is the graph of a polynomial function because it is smooth and continuous. It is the graph of a cubic function.

Characteristics of Polynomial Graphs
We know that a Table of Values (negative, zero and positive values of x to determine the values of y) is required to draw the graph of any function (including polynomial functions).
However, there are some interesting characteristics of the graphs of polynomial functions.
These characteristics help us visualize how the graph of any polynomial will look like even without the Table of Values.
These characteristics include:
(I.) Intercepts.
(II.) Multiplicity of Zeros (x-intercept touching or crossing the x-axis).
(III.) Turning Points.
(IV.) End Behavior.
(V.) Symmetry.

Intercepts
Intercepts are points where the graph touches/crosses the axis.
NOTE: A point has two coordinates: the x-coordinate and the y-coordinate.

(1.) x-intercept is the point where the graph touches/crosses the x-axis
To find the x-intercept:
Set y = 0
and
Solve for x
x-intercept = (x-value, 0)

(2.) y-intercept is the point where the graph crosses the y-axis
To find the y-intercept:
Set x = 0
and
Solve for y
y-intercept = (0, y-value)

Multiplicity of Zeros
(1.) The zero of a polynomial is also known as the root or solution
It is the value of the independent variable (x-value) for which the value of the dependent variable (y-value) is zero.

(2.) Given a polynomial: $y = f(x)$:
To find the zero(s):
Set $f(x) = 0$ and Solve for $x$

Exercise 2:
Review the definitions/concepts of x-intercept and zero of a polynomial graph.
Are they the same?
If yes, state so.
If no, state the difference.

Answer: (Click to Show/Hide)

There is a little difference.
The x-intercept of a polynomial graph is the point for which $f(x)$ is zero.
A point has two coordinates: the x-coordinate and the y-coordinate.

The zero of a polynomial graph is the value for which $f(x)$ is zero.
The value is the x-coordinate of the x-intercept.
This implies that the zero of a polynomial graph is the x-coordinate of the x-intercept of the graph.

(3.) Given a zero of a polynomial, say k:
x = k is the zero
x − k is the factor
Similarly:
Given a factor of a polynomial, say x − k:
x = k is the zero
This means that: we need to set the factor to zero to find the zero.
I hope you are not confused by the term: zero.
If you feel confused, use "solution" rather than "zero".
This is what I mean.
This means that: we need to set the factor to zero to find the solution.

(4.) The multiplicity of a zero of a polynomial is the number of occurrence of it's zero.
It can also be defined as the number of times it's factor occurs in the polynomial when the polynomial is in factored form.
It tells us how the graph behaves (touching or crossing the x-axis) at that zero.
If k is a zero of even multiplicity, then the graph touches the x-axis and turns around at k.
If k is a zero of odd multiplicity, then the graph crosses the x-axis at k.
Mnemonic:
even — touch
odd — cross

Example 3:

As seen from the graph:

$ f(x) = (x - 3)^2 \\[3ex] f(x) = (x - 3)(x - 3) \\[3ex] factor:\;\; x - 3 \\[3ex] zero:\;\; x = 3 \\[3ex] multiplicity:\;\; 2 \\[3ex] $ Because the zero: x = 3 has an even multiplicity (2 is an even number), the graph touches the x-axis at that zero.

$ g(x) = (x + 5)^3 \\[3ex] y = (x + 5)(x + 5)(x + 5) \\[3ex] factor:\;\; x + 5 \\[3ex] zero:\;\; x = -5 \\[3ex] multiplicity:\;\; 3 \\[3ex] $ Because the zero: x = −5 has an odd multiplicity (3 is an odd number), the graph crosses the x-axis at that zero.

Now, here's one for you to do.
Exercise 3:
Turning Points
Turning Points are points at which the graph changes direction from increasing to decreasing or from decreasing to increasing.
(1.) Given a polynomial say $f(x)$ of degree say n, the graph of $f(x)$ has at most n − 1 turning points.
In other words, the number of turning points of a polynomial of degree n is at most n − 1.
In other words, the maximum number of turning points of a polynomial of degree n is n − 1.

(2.) Similarly, given a polynomial say $f(x)$ that has n − 1 turning points, the graph of $f(x)$ has at least degree n
In other words, the degree of a polynomial that has n − 1 turning points, is at least n
In other words, the minimum degree of a polynomial with n − 1 turning points is n

Example 4:
The graph of $y = 5x^6 - 3x^4 + 2x - 9$ has at most how many turning points?
The polynomial is of degree 6
Hence, the graph has at most 5 (6 − 1) turning points.

Given the general/standard form of a polynomial function: $f(x) = a_nx^n + a_{n - 1}x^{n - 1} + a_{n - 2}x^{n - 2} + ... + a_1x + a_0$ resembles the graph of the power function: $f(x) = a_nx_n$ for large values of |x|
In other words, for large values of |x|, the graph of a polynomial function resembles the graph of it's leading term.
For example:
The graph of the function $f(x) = 3x^4 - x^3 + 4x^2 - 7x - 8$ resembles the graph of $y = 3x^4$ for large values of $|x|$
This knowledge leads us to:

End Behavior
We can tell how a graph behaves at its ends.
In other words: As x increases without bounds, how does y behave? (Behavior on the right)
As x decreases without bounds, how does y behave? (Behavior on the left)
End behavior of a polynomial function describes how the graph of that polynomial behaves at both ends (right end and left end).
Alternatively, we can define End Behavior of a polynomial as the behavior of the dependent variable (y) as the value of the independent variables (x) increases or decreases without bounds.

For those of you who are good at memorization, you are welcome to memorize.
However, I think it is much better to understand than memorize.
So, I do not just want to give you a Table that summarizes end bahavior of polynomial functions.
Let us draw Table of Values for each scenario that will be included in the Summary Table.

Assume four polynomial functions (two parent functions and their vertical reflections):
$ y = x \\[3ex] y = -x \\[3ex] y = x^2 \\[3ex] y = -x^2 \\[3ex] $

As we can see from this table:
(1.) As x decreases without bounds (from −1 to −2 to −∞):
the graph, y also decreases without bounds (from −1 to −2 to −∞)
This implies that: As $x \rightarrow -\infty$, $y \rightarrow -\infty$
This implies that: the graph is falling on the left
This implies that: the graph is down on the left

(2.) As x increases without bounds (from 1 to 2 to ∞):
the graph, y also increases without bounds (from 1 to 2 to ∞)
This implies that: As $x \rightarrow \infty$, $y \rightarrow \infty$
This implies that: the graph is rising on the right
This implies that: the graph is up on the right

(3.) Therefore, the end behavior is:

As $x \rightarrow -\infty$, $y \rightarrow -\infty$
As $x \rightarrow \infty$, $y \rightarrow \infty$
or
down on the left/up on the right
or
down/up

(4.) This end bahavior applies to any polynomial in standard form: $f(x) = ax^n + ...$ where:
a is any positive number.
n is any positive odd number.

(5.) This implies that the end behavior of the applicable polynomial will resemble the end behavior of $y = x$
fall/down on the left; rise/up on the right


Show students several examples using the GeoGebra graphing software.

As we can see from this table:
(1.) As x decreases without bounds (from −1 to −2 to −∞):
the graph, y increases without bounds (from 1 to 2 to ∞)
This implies that: As $x \rightarrow -\infty$, $y \rightarrow \infty$
This implies that: the graph is rising on the left
This implies that: the graph is up on the left

(2.) As x increases without bounds (from 1 to 2 to ∞):
the graph, y decreases without bounds (from −1 to −2 to −∞)
This implies that: As $x \rightarrow \infty$, $y \rightarrow -\infty$
This implies that: the graph is falling on the right
This implies that: the graph is down on the right

(3.) Therefore, the end behavior is:

As $x \rightarrow -\infty$, $y \rightarrow \infty$
As $x \rightarrow \infty$, $y \rightarrow -\infty$
or
up on the left/down on the right
or
up/down

(4.) This end bahavior applies to any polynomial in standard form: $f(x) = ax^n + ...$ where:
a is any negative number.
n is any positive odd number.
This means that for the polynomial:

$ f(x) = -3x^5 + x^3 - 6x^2 + 9 \\[3ex] \underline{End\;\;Behavior} \\[3ex] f(x) \rightarrow \infty \;\;as\;\; x \rightarrow -\infty \\[3ex] f(x) \rightarrow -\infty \;\;as\;\; x \rightarrow \infty \\[3ex] $ (5.) This implies that the end behavior of the applicable polynomial will resemble the end behavior of $y = -x$
rise/up on the left; fall/down on the right


Show students several examples using the GeoGebra graphing software.

As we can see from this table:
(1.) As x decreases without bounds (from −1 to −2 to −∞):
the graph, y increases without bounds (from 1 to 4 to ∞)
This implies that: As $x \rightarrow -\infty$, $y \rightarrow \infty$
This implies that: the graph is rising on the left
This implies that: the graph is up on the left

(2.) As x increases without bounds (from 1 to 2 to ∞):
the graph, y also increases without bounds (from 1 to 4 to ∞)
This implies that: As $x \rightarrow \infty$, $y \rightarrow \infty$
This implies that: the graph is rising on the right
This implies that: the graph is up on the right

(3.) Therefore, the end behavior is:

As $x \rightarrow -\infty$, $y \rightarrow \infty$
As $x \rightarrow \infty$, $y \rightarrow \infty$
or
up on the left/up on the right
or
up/up

(4.) This end bahavior applies to any polynomial in standard form: $f(x) = ax^n + ...$ where:
a is any positive number.
n is any positive even number.

(5.) This implies that the end behavior of the applicable polynomial will resemble the end behavior of $y = x^2$
rise/up on the left; rise/up on the right


Show students several examples using the GeoGebra graphing software.

As we can see from this table:
(1.) As x decreases without bounds (from −1 to −2 to −∞):
the graph, y also decreases without bounds (from −1 to −4 to −∞)
This implies that: As $x \rightarrow -\infty$, $y \rightarrow -\infty$
This implies that: the graph is falling on the left
This implies that: the graph is down on the left

(2.) As x increases without bounds (from 1 to 2 to ∞):
the graph, y decreases without bounds (from −1 to −4 to −∞)
This implies that: As $x \rightarrow \infty$, $y \rightarrow -\infty$
This implies that: the graph is falling on the right
This implies that: the graph is down on the right

(3.) Therefore, the end behavior is:

As $x \rightarrow -\infty$, $y \rightarrow -\infty$
As $x \rightarrow \infty$, $y \rightarrow -\infty$
or
down on the left/down on the right
or
down/down

(4.) This end bahavior applies to any polynomial in standard form: $f(x) = ax^n + ...$ where:
a is any negative number.
n is any positive even number.

(5.) This implies that the end behavior of the applicable polynomial will resemble the end behavior of $y = -x^2$
fall/down on the left; fall/down on the right


Show students several examples using the GeoGebra graphing software.

In summary, the information can be displayed in the table.

Summary of End Behavior of Polynomial Graph

a is the leading coefficient n is the degree	n is positive odd	n is positive even
a > 0	As $x \rightarrow -\infty$, $y \rightarrow -\infty$ As $x \rightarrow \infty$, $y \rightarrow \infty$ fall on the left/rise on the right down on the left/up on the right fall/rise down/up	As $x \rightarrow -\infty$, $y \rightarrow \infty$ As $x \rightarrow \infty$, $y \rightarrow \infty$ rise on the left/rise on the right up on the left/up on the right rise/rise up/up
a < 0	As $x \rightarrow -\infty$, $y \rightarrow \infty$ As $x \rightarrow \infty$, $y \rightarrow -\infty$ rise on the left/fall on the right up on the left/down on the right rise/fall up/down	As $x \rightarrow -\infty$, $y \rightarrow -\infty$ As $x \rightarrow \infty$, $y \rightarrow -\infty$ fall on the left/fall on the right down on the left/down on the right fall/fall down/down

Symmetry
Polynomials (power functions) of the form $f(x) = x^n$, where n is an even integer, are even functions.
Examples include: $x^2,\;\;x^4,\;\;x^6$ among others.
Hence, they are symmetric with respect to the y-axis.
Their graphs always contain the points: (−1, 1), (0, 0), and (1, 1).
This is because:

$ f(x) = x^n \\[3ex] Assume\;\; n = 2 \\[3ex] f(x) = x^2...quadratic function \\[3ex] For\;\;x = -1 \\[3ex] f(-1) = (-1)^2 \\[3ex] f(-1) = 1 \\[5ex] (-1, 1) \\[3ex] For\;\;x = 0 \\[3ex] f(0) = (0)^2 \\[3ex] f(0) = 0 \\[3ex] (0, 0) \\[5ex] For\;\;x = 1 \\[3ex] f(1) = (1)^2 \\[3ex] f(1) = 1 \\[3ex] (1, 1) \\[3ex] $ These points: (−1, 1), (0, 0), and (1, 1) also apply for quartic, hexic, and all even-ordered functions.

Polynomials (power functions) of the form $f(x) = x^n$, where n is an odd integer, are odd functions.
Examples include: $x,\;\;x^3,\;\;x^5$ among others.
Hence, they are symmetric with respect to the origin.
Their graphs always contain the points (−1, −1), (0, 0), and (1,1).

$ f(x) = x^n \\[3ex] Assume\;\; n = 1 \\[3ex] f(x) = x^1...linear function \\[3ex] For\;\;x = -1 \\[3ex] f(-1) = (-1)^1 \\[3ex] f(-1) = -1 \\[5ex] (-1, -1) \\[3ex] For\;\;x = 0 \\[3ex] f(0) = (0)^1 \\[3ex] f(0) = 0 \\[3ex] (0, 0) \\[5ex] For\;\;x = 1 \\[3ex] f(1) = (1)^1 \\[3ex] f(1) = 1 \\[3ex] (1, 1) \\[3ex] $ These points: (−1, −1), (0, 0), and (1, 1) also apply for cubic, pentic, and all odd-ordered functions.

The theorems on polynomials are:

(1.) Factor Theorem states that a polynomial say f(x) has a factor (x−c) if and only if f(c) = 0

$ Polynomial:\;\; f(x) \\[3ex] Factor:\;\;x - c \\[3ex] Zero:\;\; x - c = 0 \\[3ex] x = c \\[3ex] $ Example: If $f(x) = x^3 + 8x^2 + 7x - 10$ has a factor $x + 2$, illustrate the Factor Theorem.

$ f(x) = x^3 + 8x^2 + 7x - 10 \\[3ex] Factor:\;\; x + 2 \\[3ex] Zero:\;\; x + 2 = 0 \\[3ex] x = -2 \\[3ex] \underline{Factor\;\;Theorem} \\[3ex] f(-2) = (-2)^3 + 8(-2)^2 + 7(-2) - 10 \\[3ex] = - 8 + 32 - 14 - 10 \\[3ex] = 0 \\[3ex] $ (2.) Remainder Theorem

(3.) Given a quadratic polynomial equation in standard form, say f(x) = 0:
and two other polynomials, say: the first polynomial: y = k(x) and the second polynomial: y = p(x):
(a.) The solution of the polynomial equation is the intersection of the graphs of the two other polynomials.
(b.) The solution of polynomial equation is the difference between the second polynomial and the first polynomial.
In other words:

$ solution\;\;of\;\;polynomial\;\;equation = second\;\;polynomial - first\;\;polynomial \\[3ex] f(x) = p(x) - k(x) $