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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

Samdom For Peace


I greet you this day,
First: read the notes. Second: view the videos. Third: solve the questions/solved examples. Fourth: check your solutions with my thoroughly-explained solutions. Fifth: check your answers with the calculators as applicable.
I wrote the codes for some of the calculators using JavaScript, a client-side scripting language. Please use the latest internet browsers.
The Wolfram Alpha widgets (many thanks to the developers) was used for the rest of 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.

(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.


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

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.

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.

$$3 + 7 = 10$$ $$3 = augend$$ $$7 = addend$$ $$10 = sum$$

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.

$$3 - 7 = -4$$ $$3 = minuend$$ $$7 = subtrahend$$ $$-4 = difference$$

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.

$$3 * 10 = 30$$ $$3 = multiplier$$ $$10 = multiplicand$$ $$30 = product$$

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.

$$12 \div 7 = 1 \:R\: 5$$ $$12 = dividend$$ $$10 = divisor$$ $$1 = quotient$$ $$5 = remainder$$

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:
that is a combination of constants and/or variables,
and in which the variable(s) do not have negative exponents or fractional exponents.

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.

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.
A polynomial that has five terms is known as a quintinomial.

Introduction

Please note:
added to
subtracted from
multiplied by
divided by

Add and Subtract Polynomials


Factoring Formulas

Difference of Two Squares
$x^2 - y^2 = (x + y)(x - y)$


Difference of Two Cubes
$x^3 - y^3 = (x - y)(x^2 + xy - y^2)$


Sum of Two Cubes
$x^3 + y^3 = (x + y)(x^2 - xy + y^2)$

Simplify Polynomial Functions Calculator

This calculator will:
(1.) Simplify polynomial functions.

To use the calculator, please:
(1.) Type the function/expression in the textbox (the bigger textbox).
(2.) Type it according to the examples I listed.
(3.) Delete the "default" expression/function in the textbox of the calculator.
(4.) Copy and paste the expression/function you typed, into the small textbox of the calculator.
(5.) Click the "Submit" button.
(6.) Check to make sure that it is the correct expression you typed.
(7.) Review the answers. At least one of the answers is probably what you need.

Solve

Domain and Range of Polynomial Functions Calculator

This calculator will:
(1.) Determine the domain of a function.
(2.) Determine the range of a function.
(3.) Write the domain of the function in set notation.
(4.) Write the range of the function in set notation.
(5.) Graph the domain on a number line.
(6.) Graph the range on a number line.

To use the calculator, please:
(1.) Type your function (equation) or expression in the textbox (the bigger textbox).
(2.) Type it according to the examples I listed.
(3.) Copy and paste the function (equation) you typed, into the small textbox of the calculator.
(4.) Click the "Submit" button.
(5.) Check to make sure it is the correct function or expression you typed.
(6.) Review the answer.

Function:

Evaluate Polynomial Functions Calculator

This calculator will:
(1.) Evaluate a function for a specified value.
(2.) Return the answer in the simplest form.
(3.) Graph the function and indicate the specified value.

To use the calculator, please:
(1.) Assume the function is $f(x)$.
(2.) Type your expression in the first textbox - bigger Textbox 1.
(3.) Type your specified value in the second textbox - bigger Textbox 2.
(4.) Type them according to the examples I listed.
(5.) Delete the default expression in the first textbox of the calculator.
(6.) Delete the default value in the second textbox of the calculator.
(7.) Copy and paste the expression you typed, into the first textbox of the calculator.
(8.) Copy and paste the specified value you typed, into the second textbox of the calculator.
(9.) Click the "Submit" button.
(10.) Check to make sure that the expression and specified value are your questions.
(11.) Review the answer(s). At least one of the answers is what you need.

Function:

Specified Value:

Arithmetic Operations on Polynomial Functions Calculator

This calculator will:
(1.) Add two functions.
(2.) Subtract two functions.
(3.) Multiply two functions.
(4.) Not Divide two functions. Use the Long Division Calculator.
(5.) Calculate the result of a function raised to an exponent value.
(6.) Perform arithmetic operations on functions according to the order of operations.
(7.) Graph the result of the arithmetic operation.

To use the calculator, please:
(1.) Assume the first function is $f(x)$.
(2.) Assume the second function is $g(x)$.
(3.) Type the first function in the first textbox - bigger Textbox 1.
(4.) Type the second function in the first textbox - bigger Textbox 2.
(5.) Type the arithmetic operation(s) in the third textbox - bigger Textbox 3.
(6.) Type them according to the examples I listed.
(7.) Copy and paste the first expression you typed, into the first textbox of the calculator.
(8.) Copy and paste the second expression you typed, into the second textbox of the calculator.
(9.) Copy and paste the arithmetic operation(s) you typed, into the third textbox of the calculator.
(10.) Click the "Submit" button.
(11.) Check to make sure that the expressions you typed are the actual expressions of your question.
(12.) Review all the answer(s). At least one of the answers is what you need.

$f(x) = $

$g(x) = $

$Arithmetic\: Operation:$

Division of Polynomial Functions Calculator

This calculator will:
(1.) Divide any two real polynomial functions.
(2.) Divide any two complex polynomial functions.
(3.) Divide any two real/complex polynomial functions.
(4.) Express the result in the form of the Division Algorithm.
This means that it will give the dividend, divisor, quotient, and the remainder.
(5.) Display the graph of the result.
(6.) Give the domain in set notation.
(7.) Give the range in set notation.

To use the calculator, please:
(1.) Know that the first function is the dividend. $Dividend$ for the bigger Textbox 1 is the $divide$ for the calculator.
(2.) Know that the second function is the divisor. $Divisor$ for the bigger Textbox 2 is the $by$ for the calculator.
(3.) Type the dividend in the first textbox - bigger Textbox 1.
(4.) Type the divisor in the first textbox - bigger Textbox 2.
(5.) Type them according to the examples I listed.
(6.) Copy and paste the dividend you typed, into the first textbox of the calculator.
(7.) Copy and paste the divisor you typed, into the second textbox of the calculator.
(8.) Click the "Submit" button.
(9.) Check to make sure that the expressions you typed are the actual expressions of your question.
(10.) Review all the answer(s). At least one of the answers is what you need.

$Dividend = $

$Divisor = $

Factor Polynomial Functions Calculator

This calculator will:
(1.) Factor any polynomial function.
(2.) Return the answer in factored form.

To use the calculator, please:
(1.) Type your function/expression in the textbox (the bigger textbox).
(2.) Type it according to the examples I listed.
(3.) Delete the "default" function/expression in the textbox of the calculator.
(4.) Copy and paste the expression you typed, into the small textbox of the calculator.
(5.) Click the "Submit" button.
(6.) Check to make sure that it is the correct function/expression you typed.
(7.) Review the answers. At least one of the answers is what you probably need.

Factor

Determine the Zeros of Functions Calculator

This calculator will:
(1.) Determine the zeros of polynomial functions.
(2.) Give the answer(s) in the simplest exact forms. (3.) Graph the solution(s) (roots/zero(s)) on a number line.
(4.) Calculate the sum of zeros as applicable.
(5.) Calculate the product of roots as applicable.
To see the answer(s) in decimals, click the "Approximate forms" link.
To see the answer(s) in the simplest / exact forms, click the "Exact forms" link.

To use the calculator, please:
(1.) Type your function/expression in the textbox (the bigger textbox).
(2.) Type it according to the examples I listed.
(3.) Delete the "default" function/expression in the textbox of the calculator.
(4.) Copy and paste the function/expression you typed, into the small textbox of the calculator.
(5.) Click the "Submit" button.
(6.) Check to make sure that it is the correct function/expression you typed.
(7.) Review the answers.

Solve

Graph Polynomial Functions Calculator

This calculator will:
(1.) Graph a polynomial function.

To use the calculator, please:
(1.) Type your function/expression in the textbox (the bigger textbox).
(2.) Type it according to the examples I listed.
(3.) Delete the "default" function/expression in the textbox of the calculator.
(4.) Copy and paste the expression you typed, into the small textbox of the calculator.
(5.) Click the "Submit" button.
(6.) Check to make sure that it is the correct function/expression you typed.
(7.) Review the graph.

Graph

Graph and Analyze Polynomial Functions Calculator

This calculator will:
(1.) Graph a polynomial function.
(2.) Analyze the graph of polynomial functions.

Graph Polynomial Functions within a Domian Calculator

This calculator will:
(1.) Graph a polynomial function within a domain.

To use the calculator, please:
(1.) Express the domain of the function in interval notation.
The first value is the lower bound. The second value is the upper bound.
(2.) Type your function/expression in the textbox (the bigger textbox).
(3.) Type the value of the lower bound. It should be either negative infinity, $-INFINITY$ or a numeric value.
This corresponds to the "from x=" field in the calculator.
(4.) Type the value of the upper bound. It should be either infinity, $INFINITY$ or a numeric value.
This corresponds to the "to" field in the calculator.
(5.) Type these according to the examples I listed.
(6.) Delete the "default" function/expression in the first textbox of the calculator.
(7.) Delete the "default" value in the second textbox of the calculator.
(8.) Delete the "default" value in the third textbox of the calculator.
(9.) Copy and paste the first function/expression you typed, into the first textbox of the calculator.
(10.) Copy and paste, or type the boundary values (lower and upper bounds) into the second and third textboxes of the calculator respectively.
(11.) Click the "Submit" button.
(12.) Check to make sure that it is the correct function/expression you typed.
(13.) Check to make sure the values that you typed are the actual values.
$-INFINITY$ should display as $-\infty$
$INFINITY$ should display as $\infty$
(14.) Review the answers.

Graph

$Lower\: Bound$

$Upper\: Bound$

Determine the Extrema of Polynomial Functions Calculator

This calculator will:
(1.) Determine the global (absolute) extrema of a polynomial function.
(2.) Determine the local (relative) extrema of a polynomial function.
(3.) Graph the function as applicable.

To use the calculator, please:
(1.) Type your function/expression in the textbox (the bigger textbox).
(2.) Type it according to the examples I listed.
(3.) Delete the "default" function/expression in the textbox of the calculator.
(4.) Copy and paste the expression you typed, into the small textbox of the calculator.
(5.) Click the "Submit" button.
(6.) Check to make sure that it is the correct function/expression you typed.
(7.) Review the answers.

Extrema of

Determine the Extrema of Polynomial Functions within a Domain Calculator

This calculator will:
(1.) Determine the global (absolute) extrema of a polynomial function within a domain as applicable.
(2.) Determine the local (relative) extrema of a polynomial function within a domain as applicable.
(3.) Graph the function as applicable.

To use the calculator, please:
(1.) Express the domain of the function as an inequality in compact form.
The first value is the lower bound. The second value is the upper bound.
(2.) Type your function/expression in the textbox (the bigger textbox).
(3.) Type the value of the lower bound. It should be either negative infinity, $-INFINITY$ or a numeric value.
(4.) Type the value of the upper bound. It should be either infinity, $INFINITY$ or a numeric value.
(5.) Type these according to the examples I listed.
(6.) Delete the "default" function/expression in the first textbox of the calculator.
(7.) Delete the "default" value in the second textbox of the calculator.
(9.) Copy and paste the first function/expression you typed, into the first textbox of the calculator.
(10.) Copy and paste the domain (expressed as an inequality in compact form) into the second textbox of the calculator.
(11.) Click the "Submit" button.
(12.) Check to make sure that it is the correct function/expression you typed.
(13.) Check to make sure the values of the domain that you typed are the actual values.
$-INFINITY$ should display as $-\infty$
$INFINITY$ should display as $\infty$
(14.) Review the answers.
NOTE: $-INFINITY$ and $INFINITY$ should not be closed. There is no smallest number in this world, neither is there a largest number.

$f(x):$

$Domain:$


References

Chukwuemeka, S.D (2016, April 30). Samuel Chukwuemeka Tutorials - Math, Science, and Technology. Retrieved from https://www.chukwuemekasamuel.com

Coburn, J., & Coffelt, J. (2014). College Algebra Essentials ($3^{rd}$ ed.). New York: Mc-Graw Hill

Bittinger, M. L., Beecher, J. A., Ellenbogen, D. J., & Penna, J. A. (2017). Algebra and Trigonometry: Graphs and Models ($6^{th}$ ed.). Boston: Pearson.

Sullivan, M., & Sullivan, M. (2017). Algebra & Trigonometry ($7^{th}$ ed.). Boston: Pearson.

Alpha Widgets Overview Tour Gallery Sign In. (n.d.). Retrieved from http://www.wolframalpha.com/widgets/