Polynomials
Polynomials
Definition
A polynomial is a single term or a sum
or difference of terms in which all variables have whole-number exponents and no variable appears in the
denominator.
Each term can be either a constant, a
variable, or a combination of coefficients and variables.
The numerical part of the term is the coefficient.
The highest power is the degree of the
polynomial.
Not a polynomial: Polynomial:
Incorrect:
The coefficient of the term is 6. The degree of the polynomial
is 3.
Correct:
The coefficient of the term is . The degree of the polynomial
is 5. 3 4 5 3 6 2 3 2 + − + + − y y y x x
3
4
3
6
2 2 3+
−
+
y
y
x
x
xy 6 − 6 − 2 4 6 3x3 − x5 − x +Common Mistakes
xy 6 − 2 4 6 3x3 − x5 − x+Complete Manual: ..\Polynomial Review.docx
Polynomials
Types of Polynomials
Monomial-A constant, or the
product of a constant, and one or more variables raised to a whole number.
Example:
Polynomial-Any finite sum (or difference) of terms. Example:
z
y
x
2 36
−
3 2 2 32
9
3
4
x
y
−
z
+
x
y
−
xz
Binomial-A polynomial consisting of exactly two terms.
Example:
Trinomial-A polynomial consisting of exactly three terms.
Example:
7
2
x
−
4
3− x
+
x
Polynomials-Adding/Subtracting
How to Add and Subtract
Polynomials
Common Mistakes
To add or subtract polynomials combine like terms (group
together the same variable terms with the same degrees).
When subtracting, if the
subtraction sign (or negative sign) is outside of a parenthesis, you must distribute the negative sign to each of the terms inside the parenthesis. Addition Incorrect: Correct: Subtraction Incorrect: Correct: ) 6 2 2 ( ) 5 4 3 ( : Simplify x2 − x+ + − x2 + x− ) 6 2 2 ( ) 5 4 3 ( : Simplify x2 − x+ − − x2 + x− 1 or 1 2 2 6 4 − − − − x x x 1 2 2 − x− x 6 2 2 5 4 3x2 − x+ + x2 + x− 11 6 5 6 2 2 5 4 3 2 2 2 + − = + − + + − x x x x x x
Complete Manual: ..\Polynomial Review.docx
Polynomials-Multiplying
How to Multiply
Polynomials
Common Mistakes
To multiply a monomial by a
monomial multiply the coefficients together then multiply the
variables using the same rules that apply as with exponents.
To multiply a monomial and a polynomial distribute the
monomial across the polynomial. Follow the same rules as with multiplying monomials
Multiply monomials
Incorrect:
Correct:
Multiply monomial and polynomial
Incorrect: Correct: ) 2 )( 3 ( : Simplify − x2y4 − x3y3 ) 6 2 2 ( 3 : Simplify − x2 − x3 + x− 12 6 6x y 7 5 6x y 2 2 6 2 18 6 6 or 6 2 6x + x− x − x + x 2 3 5 18 6 6x − x + x
Polynomials-Multiplying (continued)
How to Multiply
Polynomials
Common Mistakes
To multiply binomial by a binomial multiply each term in the first
binomial by each term in the second binomial. Use “FOIL” method to
assist in remembering which terms need to be multiplied with which. Combine like terms.
To multiply a polynomial by another polynomial multiply each term in the first polynomial by each term in the second polynomial. Combine like terms. Multiply binomials Incorrect: Correct: Multiply polynomials Incorrect: Correct: ) 2 3 )( 5 2 ( : Simplify x− x+ ) 4 5 3 )( 2 ( : Simplify x+ x3 + x− 10 15 4 6 or 10 6x2 − x2 + x− x− 4 ) 5 ( 2 ) 3 ( x2 + x − x 8 6 5 6 3 8 10 6 4 5 3 ) 4 ( 2 ) 5 ( 2 ) 3 ( 2 ) 4 ( ) 5 ( ) 3 ( 2 3 4 3 2 4 3 3 − + + + = − + + − + = − + + − + x x x x x x x x x x x x x x x x 10 11 6 10 15 4 6x2 + x− x− = x2 − x−
Complete Manual: ..\Polynomial Review.docx
Polynomials-Special Products
How to Multiply Special
Products
Common Mistakes
Multiply Incorrect: Correct: Multiply Incorrect: Correct: 2 ) 5 3 ( : Simplify x+ 3 ) 3 2 ( : Simplify x+ 25 3 or 25 9x2 + x2 + 25 30 9 ) 5 ( ) 5 )( 3 ( 2 ) 3 ( 2 2 2 + + = + + x x x x 3 2 2 3 3 3 2 2 3 3 2 2 2 2 2 2 2 2
3
3
)
(
3
3
)
(
)
)(
(
2
)
(
2
)
(
b
ab
b
a
a
b
a
b
ab
b
a
a
b
a
b
a
b
a
b
a
b
ab
a
b
a
b
ab
a
b
a
−
+
−
=
−
+
+
+
=
+
−
=
−
+
+
−
=
−
+
+
=
+
2 2 ) ( or ) (a+b a−b 3 3 ) ( or ) (a+b a−b 27 2 or 27 8x2 + x3 + ) 3 ( ) 3 )( 2 ( 3 ) 3 ( ) 2 ( 3 ) 2 ( x 3 + x 2 + x 2 + 3Polynomials-Dividing
How to Divide Polynomials
Common Mistakes
To divide a polynomial by a single term treat the division as a
simplification and reduce each term to the lowest terms possible.
Divide by a monomial Incorrect: Correct: x x x 4 4 2 : Simplify 2 + 2 2 2 2 2 4 4 2 or 1 2 4 4 2 x x x x x x x x ≠ + + ≠ + 1 2 2 2 2 2 2 ) 2 ( 2 or 1 2 4 4 4 2 4 4 2 2 2 + = + = / / × + / / + = + = + x x x x x x x x x x x x x
Complete Manual: ..\Polynomial Review.docx
Polynomials-Long Division
How to Do Long Division of
Polynomials
Common Mistakes
Long division of polynomials is the same as regular long division, with the exception that variables are included. Long Division Incorrect: Correct: . 2 by 12 6 Dividex2 − x− x+ 24 8 4 16 4 2 4 16 6 2 2 2 − − − − − + − − − + x x x x x x x x ) 16 8 ( 16 8 ) 2 ( 8 16 6 2 2 2 − − − − − + − − − − + x x x x x x x x ) 2 /( ) 24 ( ) 4 ( ) 2 ( ) 16 6 ( 2 + − + − ≠ + ÷ − − x x x x x ) 8 ( ) 2 ( ) 16 6 ( 2 − = + ÷ − − x x x x
