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

# Calculators for Polynomials

I greet you this day,
You are encouraged to solve the questions on your own first, before checking your answers with the calculators.
You may need to refresh your browser after each calculation.
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 do, please be positive.
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

## Evaluate Polynomials

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.

(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.

• Using the Evaluate Polynomial Functions Calculator
• Type: $3x^2 - 3x + 1$ as 3 * x^2 - 3 * x + 1
$g(0)$ means x = 0
$g(-2)$ means x = -2
$g(3)$ means x = 3
$g(-x)$ means x = -x
$g(3y)$ means x = 3y
$g(1 - t)$ means x = 1 - t
$g(7 + h)$ means x = 7 + h
• Type: $\dfrac{x - 7}{3}$ as (x - 7) / 3
$p(7)$ means x = 7
$p(-12.75)$ means x = -12.75
$p(-3)$ means x = -3
$p(x + h)$ means x = x + h
$p\left(\dfrac{2}{3} \right)$ means x = 2/3
$p(\sqrt{2})$ means x = sqrt(2)
$p(\sqrt{2})$ means x = cuberoot(2)

Function:

Specified Value:

## Domain and Range of Polynomials

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.

(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.

• Using the Domain and Range Calculator
• Type: $p(x) = 2x - 7$ as p(x) = 2 * x - 7
• Type: $p(x) = 3x^3 - 2x^2 + 7x - 5$ as p(x) = 3 * x^3 - 2 * x^2 + 7 * x - 5
• Type: $f(x) = 12 - 3x^7$ as f(x) = 12 - 3 * x^7
• Type: $g(x) = -3(x - 5)^3(2x - 9)^5$ as g(x) = -3 * (x - 5)^3 * (2 * x - 9)^5
• Type: $p(x) = -3x^2(5 - 2x)^3(12 + x)^5$ as p(x) = -3 * x^2 * (5 - 2 * x)^3 * (12 + x)^5

Function:

## Simplify Polynomials

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

(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.

• Using the Simplify Polynomials Calculator
• Type: $4x + 3x$ as 4 * x + 3 * x
• Type: $5(3y - 7) - (3y + 8)$ as 5(3 * y - 7) - (3 * y + 8)
• Type: $4c + 4[5 - (d - 2)]$ as 4 * c + 4[5 - (d - 2)]
• Type: $3(-3x^2 + 2x) - (4x - 4x^2)$ as 3(-3 * x^2 + 2 * x) - (4 * x - 4 * x^2)
• Type: $5p - 4[5p - (6p^2 - 4d^3)]$ as 5 * p - 4[5 * p - (6 * p^2 - 4 * d^3)]
• Type: $3(9m^2 - 4) - [4(-2 - 3m^2) + 3]$ as 3(9 * m^2 - 4) - [4(-2 - 3 * m^2) + 3]
• Type: $\dfrac{2}{3}(2k - 9) + \dfrac{3}{4}(k + 12)$ as (2/3)(2 * k - 9) + (3/4)(k + 12)

Solve

## Arithmetic Operations on Polynomials

This calculator will:
(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.

(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.

• Using the Arithmetic Operations on Polynomial Calculator
• Type:
f(x) = $-4x + 3$ as -4 * x + 3
g(x) = $7x + 5$ as 7 * x + 5
Arithmetic Operation: $f(x) + g(x)$ as f + g
Arithmetic Operation: $f(x) - g(x)$ as f - g
Arithmetic Operation: $f(x) * g(x)$ as f * g
Arithmetic Operation: $f(x) \div g(x)$ as f / g
Arithmetic Operation: $3f(x)^2 - 7g(x)^3$ as 3 * f^2 - 7 * g^3
• Type:
f(x) = $3x^2 + 10x - 25$ as 3 * x^2 + 10 * x - 25
g(x) = $x^2 + 4x - 5$ as x^2 + 4 * x - 5
Arithmetic Operation: $f(x) + g(x)$ as f + g
Arithmetic Operation: $f(x) - g(x)$ as f - g
Arithmetic Operation: $f(x) * g(x)$ as f * g
Arithmetic Operation: $f(x) \div g(x)$ as f / g
Arithmetic Operation: $3f(x)^2 - 7g(x)^3$ as 3 * f^2 - 7 * g^3

$f(x) =$

$g(x) =$

Arithmetic Operation:

## Division of Polynomials

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.

(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.

• Using the Division of Polynomial Functions Calculator
• Type:
$Dividend = x^3 + 2x^2 - 30x + 144$ as x^3 + 2 * x^2 - 30 * x + 144
$Divisor = x + 8$ as x + 8
• Type:
$Dividend = 3v^4 - 16v^3 + 61v + 17$ as 3 * v^4 - 16 * v^3 + 61 * v + 17
$Divisor = v - 4$ as v - 4
• Type:
$Dividend = 6k^4 - 16k^3 + 15k^2 - 5k + 10$ as 6 * k^4 - 16 * k^3 + 15 * k^2 - 5 * k + 10
$Divisor = 3k + 1$ as 3 * k + 1
• Type:
$Dividend = d^4 - 1$ as d^4 - 1
$Divisor = d^2 - 1$ as d^2 - 1
• Type:
$Dividend = x^3 + 9ix^2 - 11ix - 8$ as x^3 + 9 * i * x^2 - 11 * i * x - 8
$Divisor = x + i$ as x + i
• Type:
$Dividend = 7x^2 + 3x - 12$ as 7 * x^2 + 3 * x - 12
$Divisor = -2x - i$ as -2 * x - i

$Dividend =$

$Divisor =$

## Factor Polynomials

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

(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.

• Using the Factor Polynomial Functions Calculator
• All outputs/answers are in the factored form.
• At least one of the answers is most likely what you need.
• Type: $12a + 8*b$ as 12 * a + 8 * b
• Type: $15p^2 + 6p$ as 15 * p^2 + 6 * p
• Type: $24x^3y^2 + 36x^2y$ as 24 * x^3 * y^2 + 36 * x^2 * y
• Type: $8c^5d^3 + 6c^4d^5 - 12c^2d^4$ as 8 * c^5 * d^3 + 6 * c^4 * d^5 - 12 * c^2 * d^4
• Type: $m(n - 1) + 7(n - 1)$ as m * (n - 1) + 7 * (n - 1)
• Type: $5x(a - b) - y(a - b)$ as 5 * x * (a - b) - y * (a - b)
• Type: $mx - 2x + my - 2y$ as m * x - 2 * x + m * y - 2 * y
• Type: $mx^2 - 2x^2 + 5m - 10$ as m * x^2 - 2 * x^2 + 5 * m - 10

Factor

## Solve Polynomial Equations (Zeros of Functions)

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 the simplest / exact forms, click the "Exact forms" link.

(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.

• Using the Solve Polynomial Equations Calculator
• All outputs/answers are written as both integers and/or decimals; and integers and/or fractions.
• Type: $f(p) = 5p - 25$ as 5 * p - 25
• Type: $g(x) = -\dfrac{1}{6} * x^3 - 3x + 7$ as (-1/6)*x^3 - 3 * x + 7
• Type: $f(x) = -4.2x^4 + x^6 + 0.1x^7$ as -4.2 * x^4 + x^6 + 0.1 * x^7
• Type: $f(x) = 6 + \dfrac{1}{6}x^4 - \dfrac{5}{7}x^3$ as 6 + (1/6) * x^4 - (5/7) * x^3
• Type: $h(x) = \sqrt{2}x^3 + 3x^2 - 2x + 2$ as sqrt(2) * x^3 + 3 * x^2 - 2 * x + 2
• Type: $f(x) = -3.476x^4 + 35.145x^3 - 97.678x^2 + 41.613x + 178.288$ as -3.476x^4 + 35.145x^3 - 97.678x^2 + 41.613x + 178.288
• Type: $f(k) = 7x^3 - x^2 - 448x + 64$ as 7*x^3 - x^2 - 448 * x + 64
• Type: $f(x) = (x + 3)^2 (x - 7)$ as (x + 3)^2 * (x - 7)
• Type: $f(x) = (x^2 - 4)^5$ as (x^2 - 4)^5
• Type: $f(x) = -3(x + 3)(x + 3) (x + 3)(x - 4)$ as -3 * (x + 3)* (x + 3) * (x + 3) * (x - 4)
• Type: $f(x) = -8(x - 4)^3(x + 3)^4x^2$ as -8 * (x - 4)^3 * (x + 3)^ 4 * x^2
• Type: $f(x) = x^7(2x - 5)^2(7 - 3x)^3$ as x^7 * (2 * x - 5)^2 * (7 - 3 * x)^3

Solve

## Graph Polynomials

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

(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.

• Using the Graph Polynomial Functions Calculator
• Type: $f(x) = 5x^2 - 25$ as 5 * x^2 - 25
• Type: $g(x) = -\dfrac{1}{6} * x^3 - 3x + 7$ as (-1/6)*x^3 - 3 * x + 7
• Type: $f(x) = -4.2x^4 + x^6 + 0.1x^7$ as -4.2 * x^4 + x^6 + 0.1 * x^7
• Type: $f(x) = 6 + \dfrac{1}{6}x^4 - \dfrac{5}{7}x^3$ as 6 + (1/6) * x^4 - (5/7) * x^3
• Type: $h(x) = \sqrt{2}x^3 + 3x^2 - 2x + 2$ as sqrt(2) * x^3 + 3 * x^2 - 2 * x + 2
• Type: $f(x) = -3.476x^4 + 35.145x^3 - 97.678x^2 + 41.613x + 178.288$ as -3.476x^4 + 35.145x^3 - 97.678x^2 + 41.613x + 178.288
• Type: $f(k) = 7x^3 - x^2 - 448x + 64$ as 7*x^3 - x^2 - 448 * x + 64
• Type: $f(x) = (x + 3)^2 (x - 7)$ as (x + 3)^2 * (x - 7)
• Type: $f(x) = (x^2 - 4)^5$ as (x^2 - 4)^5
• Type: $f(x) = -3(x + 3)(x + 3) (x + 3)(x - 4)$ as -3 * (x + 3)* (x + 3) * (x + 3) * (x - 4)
• Type: $f(x) = -8(x - 4)^3(x + 3)^4x^2$ as -8 * (x - 4)^3 * (x + 3)^ 4 * x^2
• Type: $f(x) = x^7(2x - 5)^2(7 - 3x)^3$ as x^7 * (2 * x - 5)^2 * (7 - 3 * x)^3

Graph

## Graph Polynomials Within a Domain

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

(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 $\infty$

• Using the Graph Polynomial Functions within a Domain Calculator
• Type:
Graph: $x^3 + 2x^2 - 30x + 144$ as x^3 + 2 * x^2 - 30 * x + 144
Lower Bound: $-\infty$ as -INFINITY
Upper Bound: $\infty$ as INFINITY
• Type:
Graph: $-\dfrac{1}{6} * x^3 - 3x + 7$ as (-1/6)*x^3 - 3 * x + 7
Lower Bound: $-3$ as -3
Upper Bound: $3$ as 3
• Type:
Graph: $-3(x^2 + 3)^2 (7 - 12x)$ as -3(x^2 + 3)^2 * (7 - 12x)
Lower Bound: $-7$ as -7
Upper Bound: $7$ as 7

Graph

Lower Bound:

Upper Bound:

## Extrema of Polynomials

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.

(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.

• Using the Extrema of Polynomial Functions Calculator
• Type: $f(p) = 5p^2 - 25$ as 5 * p^2 - 25
• Type: $g(x) = -\dfrac{1}{6} * x^3 - 3x + 7$ as (-1/6)*x^3 - 3 * x + 7
• Type: $f(x) = -4.2x^4 + x^6 + 0.1x^7$ as -4.2 * x^4 + x^6 + 0.1 * x^7
• Type: $f(x) = 6 + \dfrac{1}{6}x^4 - \dfrac{5}{7}x^3$ as 6 + (1/6) * x^4 - (5/7) * x^3
• Type: $h(x) = \sqrt{2}x^3 + 3x^2 - 2x + 2$ as sqrt(2) * x^3 + 3 * x^2 - 2 * x + 2
• Type: $f(x) = -3.476x^4 + 35.145x^3 - 97.678x^2 + 41.613x + 178.288$ as -3.476x^4 + 35.145x^3 - 97.678x^2 + 41.613x + 178.288
• Type: $f(k) = 7x^3 - x^2 - 448x + 64$ as 7*x^3 - x^2 - 448 * x + 64
• Type: $f(x) = (x + 3)^2 (x - 7)$ as (x + 3)^2 * (x - 7)
• Type: $f(x) = (x^2 - 4)^5$ as (x^2 - 4)^5
• Type: $f(x) = -3(x + 3)(x + 3) (x + 3)(x - 4)$ as -3 * (x + 3)* (x + 3) * (x + 3) * (x - 4)
• Type: $f(x) = -8(x - 4)^3(x + 3)^4x^2$ as -8 * (x - 4)^3 * (x + 3)^ 4 * x^2
• Type: $f(x) = x^7(2x - 5)^2(7 - 3x)^3$ as x^7 * (2 * x - 5)^2 * (7 - 3 * x)^3

Extrema of

## Extrema of Polynomials Within a Domain

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.

(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 $\infty$
NOTE: $-INFINITY$ and $INFINITY$ should not be closed. There is no smallest number in this world, neither is there a largest number.

• Using the Extrema of Polynomial Function within a Domain Calculator
• Type:
f(x): $x^3 + 2x^2 - 30x + 144$ as x^3 + 2 * x^2 - 30 * x + 144
Domain: $-\infty \lt x \lt \infty$ as -INFINITY < x < INFINITY
• Type:
f(x): $-\dfrac{1}{6} * x^3 - 3x + 7$ as (-1/6)*x^3 - 3 * x + 7
Domain: $-7 \le x \le 12$ as -7 ≤ x ≤ 12
• Type:
f(x): $-3(x^2 + 3)^2 (7 - 12x)$ as -3(x^2 + 3)^2 * (7 - 12x)
Domain: $3 \le x \lt \infty$ as 3 ≤ x < INFINITY

f(x)

Domain: