Mathematics 334 Designed by Peter Nield Notes by Katherine Daignault
Westwood High School Grade 9 Student
POLYNOMIALS
Monomial: a monomial is a single term made up of either a number, a variable, or a
product of a number and a variable.
EX:
5
5x
~ All monomials
5x 2 y 2
5 xyz 3
Degree of a Monomial
The degree of a monomial is the sum of the exponents of its variables.
EX:
5 (degree: 0)
5x (degree: 1)
2
5 x y (degree: 3)
5 x 2 yz 3 (degree: 6)
Polynomials (poly means many):
Two or more monomials linked through addition or subtraction.
EX:
3x 2 + 4 x + 2 y + 6
Binomial: a polynomial containing 2 terms.
Trinomial: a polynomial containing 3 terms.
Degree of a Polynomial
The degree of a polynomial is equal to the highest exponent.
EX:
4 x 2 + 3x (degree: 2)
LIKE-TERM MONOMIALS
- Only like term monomials can be added or subtracted.
- Monomials are like terms when they have identical variables applied to identical
bases.
3 xy & −2 xy 2
2
LIKE TERMS
2a b & − a b
2 2
LIKE TERMS
2 2
4 xy & 3 x y NOT LIKE TERMS
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Mathematics 334 Designed by Peter Nield Notes by Katherine Daignault
Westwood High School Grade 9 Student
Adding & Subtracting Polynomials
Rewrite the expression by combining the like terms within each polynomial.
EX. 1:
4 x 2 + 3x − 5
+ 2x 2 − 6x + 8
6 x 2 − 3x + 3
EX. 2:
5m 3 + 2 m 2 + 7
− 8m 3 − 6 m 2 − 4
− 3m 3 + 8m 2 + 11
EX. 3:
3x + 5
− + 4y + 8
3x − 4 y − 3
POLYNOMIAL NOTATION: P(x )
- Pronounced “P of x”
- Simply means a polynomial containing the letter x
P(x ) = 3x + 5
Q( x ) = " Q of x"
EX. 1:
* P( x ) = 3x 2 + 2 x − 5
* Q( x ) = 4 x 2 + 8
* R( x ) = 2 x + 11
P(x ) + Q( x ) =
3x 2 + 2 x − 5
+ 4x 2 + 8
7x 2 + 2x + 3
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Mathematics 334 Designed by Peter Nield Notes by Katherine Daignault
Westwood High School Grade 9 Student
Multiplying Polynomials
The Distributive Property:
MONOMIAL * POLYNOMIAL
3(4 x + 2 )
= 3 × 4x + 3 × 2
= 12 x + 6
BINOMIAL × BINOMIAL
Multiply each term in the first binomial by each term in the second.
(3x + 4)(x + 2)
= 3 x ( x + 2 ) + 4( x + 2 )
= 3x 2 + 6 x + 4 x + 8
Combine like terms
= 3 x 2 + 10 x + 8
(x + 3)(x − 4)
F.O.I.L.
Firsts (x × x ) = x 2
Outers (x × −4) = −4 x
Inners (3 × x ) = 3x
Lasts (3 × −4) = −12
Answer using F.O.I.L.:
(x + 3)(x − 4)
= x 2 − 4 x + 3x − 12
= x 2 − x − 12
EX. 1:
(a + 6)(a + 3)
= a 2 + 3a + 6a + 18
= a 2 + 9a + 18
EX. 2:
( x + 4 )2
= ( x + 4 )( x + 4)
= x 2 + 4 x + 4 x + 16
= x 2 + 8 x + 16
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Mathematics 334 Designed by Peter Nield Notes by Katherine Daignault
Westwood High School Grade 9 Student
Division of Polynomials
Dividing a polynomial by a monomial:
Divide each term in the polynomial by the monomial.
EX. 1:
(
6 x 3 + 12 x 2 − 3 x ÷ 3x )
3 2
6 x 12 x 3x
= + −
3x 3x 3x
= 2x + 4x − 1
2
EX. 2:
(24 x 4
)
y 4 − 8 x 3 y 2 + 16 x 2 y ÷ 4 xy
4 4 3 2 2
24 x y 8x y 16 x y
= − +
4 xy 4 xy 4 xy
= 6x3 y 3 − 2x 2 y + 4x
Dividing a polynomial by a polynomial:
Use long division.
EX:
x +1
x + 1 x 2 + 2x + 1
− x2 − x
x +1
− x +1
0
* EVERY POWER MUST BE REPRESENTED. IF ONE IS MISSING, ADD
IT IN WITH A COEFFICIENT OF ZERO!
EX:
(x 3
− 1) ÷ ( x − 1)
x2 + x +1
x − 1 x3 + 0x 2 + 0x − 1
− x3 − x2
x 2 + 0x
− x2 − x
x −1
− x −1
0
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