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If there is one prayer that you should pray/sing every day and every hour, it is the LORD's prayer (Our FATHER in Heaven prayer)
It is the most powerful prayer. A pure heart, a clean mind, and a clear conscience is necessary for it.
- Samuel Dominic Chukwuemeka

For in GOD we live, and move, and have our being. - Acts 17:28

The Joy of a Teacher is the Success of his Students. - Samuel Dominic Chukwuemeka

# Polynomials

I greet you this day,
Second: view the videos.
Third: solve the questions/solved examples.
Fourth: check your solutions with my thoroughly-explained solutions.

Please use the latest internet browsers. The calculators should work.
The Wolfram Alpha widgets (many thanks to the developers) is used for the calculators.
Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. You may contact me.
If you are my student, please do not contact me here. Contact me via the school's system.
Thank you for visiting.

Samuel Dominic Chukwuemeka (Samdom For Peace) B.Eng., A.A.T, M.Ed., M.S

## Objectives

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.
(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

## Vocabulary Words

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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".

## Definitions

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.

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## Introduction

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.

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?

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

The correcr answer is Option C.

$f(p) = -2p^2 + 5p + 3 \\[3ex]$ This is seen as:

$-2p^2 + 5p^1 + 3p^0 \\[3ex] Because: \\[3ex] p = p^1 ...Law\;4...Exp \\[3ex] 5p = 5p^1 \\[3ex] 1 = p^0 ...Law\;3...Exp \\[3ex] 3 = 3 * 1 = 3 * p^0 = 3p^0 \\[3ex]$ So, the exponents in this order: 2, 1, 0 are in descending order.
That makes the polynomial to be in standard form.

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.
(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.

(1.) $f(x) = 3x - 5$ Answer:
(2.) $f(p) = 3p - 5 - 7p^2$ Answer:
(3.) $-\dfrac{5}{12}y^3$ Answer:
(4.) $f(x) = -\dfrac{5}{12x^3}$ Answer:
(5.) $25k^2 - 100$ Answer:
(6.) $f(a) = -6a^5 + 12a^4 - 9a^3$ Answer:
(7.) $7$ Answer:
(8.) $6a^3 - 9a^2 - 2a + 3$ Answer:
(9.) $f(d, p) = -12p - 45d$ Answer:
(10.) $9p^2 + 27p^3 - 18p^4$ Answer:
(11.) $a^{3x} - 26a^{2x} + 156a^x - 216$ Answer:
(12.) $f(m) = 2m - 5 + \sqrt{7}m^3 - \dfrac{1}{4}m^2 + m^5$ Answer:
(13.) $f(y) = y$ Answer:
(14.) $f(y) = \dfrac{1}{y}$ Answer:
(15.) $f(x, y) = x^2 - xy + y^2$ Answer:
(16.) $f(x, y) = x^2 - xy^2 + y^2$ Answer:
(17.) $f(x, y) = x^4 - xy^2 + y^2$ Answer:
(18.) $c + d + e$ Answer:
(19.) $f(x) = 3x^{-2} + 5x^{-1} - 7$ Answer:
(20.) $f(x) = 7\sqrt{x} - 9$ Answer:
(21.) $f(x) = \sqrt{7}x - 9$ Answer:
(23.) $f(x) = 7x^7 - \pi x^6 + \dfrac{1}{4}$ Answer:

## Evaluate Polynomials

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?