Algebra: Polynomial Functions
Contents (7 topics • 68 videos)
Monomials
Introduction Videos
What is a Monomial? 
Multiplying Monomials
Dividing Monomials
Theorems and Definitions
Monomials
A monomial is a number, a variable, or the product of numbers and variables where the variables have only whole number exponents.
The degree of a monomial is the sum of its variable exponents.
Determine the degree of the monomial.
\[ -16m^3n \]
Explain why each algebraic expression is not a monomial.
\[ \frac{5}{n^3} \quad \quad 9p^3-5p^2 \quad \quad \frac{7}{3}u^5y^{-3} \]
Write a monomial with degree 8 and 2 variables.
Write a monomial with degree 4 and 3 variables.
Multiply.
\[ 5n \cdot n^3 \cdot -2 \]
Multiply.
\[ \frac{2}{7}m^5n^3 \cdot \frac{14}{5}n^2 \]
Divide. Determine if the quotient is a monomial.
\[ \frac{21k^7}{14k^6} \]
Divide. Determine if the quotient is a monomial.
\[ \frac{6^2x^3y^3}{2^3x \space \cdot \space 3y^5} \]
Calculate the area of the rectangle, including units.
\[ \text{Length: } \Big(\sqrt{3}m^2n \Big) \text{ meters } \\ \text{Width: } \Big(2\sqrt{3}mn^4 \Big) \text{ meters} \]
Documents
Practice Problems (.PDF) 
Practice Problems - Solutions (.PDF) 
Introduction to Polynomials
Introduction Videos
Introduction to Polynomials
Theorems and Definitions
Polynomials
A
polynomial is a monomial, or the sum or difference of two or more monomials.
Polynomials are classified based on their degree and number of terms.
The
degree of a polynomial is the degree of the highest degree term.
A
binomial is the sum or difference of two monomials. A
trinomial is the sum or difference of three monomials.
Following is a listing of classifications based on the degree of a polynomial:
Degree 0 - Constant
Degree 4 - Quartic
Degree 8 - Octic
Degree 1 - Linear
Degree 5 - Quintic
Degree 9 - Nonic
Degree 2 - Quadratic
Degree 6 - Sextic
Degree 10 - Decic
Degree 3 - Cubic
Degree 7 - Septic
Degree \(n\) - \(n\)th degree
A polynomial is written in
standard form if the degree of each term descends from left to right.
The
leading coefficient of a polynomial is the coefficient of its highest degree term.
Classify the polynomial based on its degree and number of terms. Then write the polynomial in standard form.
\[ p^3-3p^7+6p^5 \]
Classify the polynomial based on its degree and number of terms. Then write the polynomial in standard form.
\[ 4c-2c^3+5+7c^5-8c^2 \]
Write a polynomial in standard form that is classified as an "octic polynomial with four terms."
Write a polynomial in standard form that is classified as a "quartic binomial."
Given the polynomial, answer each part.
\[ -12x^6+7x^4-5x^2+13 \]
- What is the leading coefficient?
- What is the quartic term?
- What is the coefficient of the quadratic term?
- What is the degree of the polynomial?
Documents
Practice Problems (.PDF)
Practice Problems - Solutions (.PDF)
Adding and Subtracting Polynomials
Introduction Videos
Adding and Subtracting Polynomials
The Closure Property for Polynomials
Theorems and Definitions
Adding and Subtracting Polynomials
To add or subtract polynomials, combine like terms. Like terms are any terms that have the same variables and the same exponents on those variables. With polynomials in one variable, like terms will have the same degree.
Add. Write the sum in standard form.
\[ (9x^3+12x^2-2x)+(11x^2-10x^3+12x) \]
Add. Write the sum in standard form.
\[ (5x-8-2x^3)+(3-x^4+6x^3)+(2x^3-10x^4-x) \]
Subtract. Write the difference in standard form.
\[ (9d+12-2d^3)-(9d^3-3-d) \]
Subtract. Write the difference in standard form.
\[ (m^4+9m^3+11m^2+7m-5)-(7+8m^2-2m^4)-(-5m^3-2m-10) \]
Simplify. Write the polynomial in standard form.
\[ (11c-2c^2+5)+(3c^2-2c+3)-(6+c^2-7c) \]
Joe and Sue each have money. Sue has \( (5d-3) \) dollars and Joe has \( (45-3d) \) dollars.
- Write an algebraic expression (in terms of \(d\)) that represents the amount of money they have together.
- If \(d=8\), how much money do they have together?
- Write an algebraic expression (in terms of \(d\)) that represents how much more money Joe has than Sue.
- If \(d=5\), how much more money does Joe have than Sue?
- For what value of \(d\) will Sue and Joe have the same amount of money? How much will that be?
Documents
Practice Problems (.PDF)
Practice Problems - Solutions (.PDF)
Multiplying Polynomials
Introduction Videos
Multiplying Polynomials
Special Products with Binomials
The Closure Property for Polynomials
Theorems and Definitions
The Product Rule for Exponents
For any real number \(x\) and natural numbers \(m\) and \(n\):
\[ x^m \cdot x^n = x^{m+n} \]
The Square of a Binomial
The square of a binomial has two formulas, based on whether the binomial is a sum or a difference.
\[ (a+b)^2=a^2+2ab+b^2 \]
\[ (a-b)^2=a^2-2ab+b^2 \]
The Difference of Squares
The product of the sum and differenece of the same two terms is equal to the difference of the squares of the terms.
\[ (a-b)(a+b)=a^2+ab-ab-b^2=a^2-b^2 \]
Multiply.
\[ -2x^3(5x^2-2x+9) \]
Multiply.
\[ (7n-4)(6n+3) \]
Multiply.
\[ (r^3-5t^2)(5r^2+3t) \]
Multiply.
\[ (9c^3-2d)(9c^3+2d) \]
Multiply.
\[ (2w+3)^2 \]
Multiply.
\[ (5p^2+8p-2)(7p^2-p+4) \]
Multiply.
\[ (2x^5-5x^3+x)(3x^4+5x^2-2) \]
Multiply.
\[ (5y^3-8y+3)^2 \]
Multiply.
\[ (2a-5)(2a+5)(5a-2) \]
Multiply.
\[ (m+1)^2(m-2)^2 \]
Multiply.
\[ (2x-1)^4 \]
A certain rectangular prism has a length of \( (5a-8) \) inches, a width of \( (a^2+9) \) inches, and a height of \( (3a+8) \) inches.
- Write an algebraic expression (in terms of \(a\)) that represents the volume of the rectangular prism, including units.
- If \(a=6\), determine the dimensions of the rectangular prism, including units.
- If \(a=6\), determine the volume of the rectangular prism, including units.
Documents
Practice Problems (.PDF)
Practice Problems - Solutions (.PDF)
Dividing a Polynomial by a Monomial
Introduction Videos
Dividing a Polynomial by a Monomial
The Closure Property for Polynomials
Theorems and Definitions
The Quotient Rule for Exponents
For any real number \(x\) and natural numbers \(m\) and \(n\):
\[ \frac{x^m}{x^n} = x^{m-n} = \frac{1}{x^{n-m}} \]
Dividing a Polynomial by a Monomial
To divide a polynomial by a monomial, break the fraction into separate fractions, all with the common denominator, then simplify each fraction independently using the following property:
\[ \frac{a+b-c}{d} = \frac{a}{d}+\frac{b}{d}-\frac{c}{d} \]
Divide.
\[ \frac{30a^3+10a^2+3a}{10a} \]
Divide.
\[ \frac{3v^5+4v^3-2v}{-6v} \]
Divide.
\[ \frac{32n^3+2n^2+4}{-8n^3} \]
Divide.
\[ \frac{(x+3)(2x-5)+3x-5}{2x} \]
A certain rectangular prism has a volume of \( (45x^4+75x^3) \) cubic meters. If the width of the prism is \( (3x) \) meters and the depth of the prism is \( (5x^2) \) meters, then determine its height, including units, in terms of \(x\).
Documents
Practice Problems (.PDF)
Practice Problems - Solutions (.PDF)
Long Division with Polynomials
Introduction Videos
Long Division with Polynomials
Dividing a Polynomial by a Quadratic
Polynomial Long Division with Missing Terms
Theorems and Definitions
The Division Algorithm
If \(a(x)\) and \(b(x)\) are polynomials where \(b(x) \ne 0\), and the degree of \(a(x)\) is greater than the degree of \(b(x)\), then there exist unique polynomials \(q(x)\) and \(r(x)\) such that
\[ a(x) = b(x) \cdot q(x) + r(x) \]
where \(r(x)=0\) or the degree of \(r(x)\) is less than the degree of \(b(x)\). If the remainder \(r(x)\) is zero, then \(b(x)\) divides evenly into \(a(x)\).
Divide.
\[ (7x^3+23x^2-9x+44) \div (x+4) \]
Divide.
\[ (2x^5+5x^4-6x^3-15x^2+10x+29) \div (2x+5) \]
Divide.
\[ (x^5-9x^3-2x^2+10x+19) \div (x^2-8) \]
Divide.
\[ (33x^4+38x^3-9x^2+32x-14) \div (3x^2+4x-2) \]
Documents
Practice Problems (.PDF)
Practice Problems - Solutions (.PDF)
Synthetic Division with Polynomials
Introduction Videos
Synthetic Division with Polynomials
Synthetic Division with Missing Terms
An Extended Version of Synthetic Division with Polynomials
Writing Polynomials in Quotient/Remainder Form
Divide using synthetic division.
\[ (3x^5-9x^4+8x^3-24x^2-3x+9) \div (x-3) \]
Divide using synthetic division.
\[ (x^7+5x^6-x^4-5x^3-12x-60) \div (x+5) \]
Divide using synthetic division.
\[ (8x^5-56x^4+5x^2-35x+17) \div (x-7) \]
Divide using synthetic division.
\[ \bigg(x^3+\Big(1-\sqrt{2}\Big) x^2+ \Big(-12-\sqrt{2} \Big)x+12\sqrt{2} \bigg) \div \Big(x-\sqrt{2} \Big) \]
Divide using synthetic division.
\[ \bigg(3x^3+ \Big(13-3\sqrt{3} \Big) x^2+ \Big(13-7\sqrt{3} \Big)x+\sqrt{3}-2 \bigg) \div \Big(x+2-\sqrt{3} \Big) \]
Divide using the extended version of synthetic division.
\[ (10x^4-6x^3-40x^2+59x-21) \div (5x-3) \]
Divide using the extended version of synthetic division.
\[ (16x^3+42x^2-45x+17) \div (2x+7) \]
Write the function in the form \( f(x)=(x-k) \cdot q(x) + r \) for the given value of \( k \).
\[ f(x)=5x^4-2x^3-10x^2+16x-4, \quad k=\frac{2}{5} \]
Write the function in the form \( f(x)=(x-k) \cdot q(x) + r \) for the given value of \( k \).
\[ f(x)=3x^3-6x^2-15x+9\sqrt{5}-24, \quad k=2-\sqrt{5} \]
Documents
Practice Problems (.PDF)
Practice Problems - Solutions (.PDF)
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