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Multiplying Polynomials
Multiplying a Polynomial by a Monomial
To multiply a polynomial by a monomial we use a Distributive Property as well as the rule
for multiplying exponential expressions.
EXAMPLE: 4 x^2 ( x + 8)
We will first multiply 4 x^2 and x. Then we will multiply 4 x^2 and 8.
4 x^2 ( x ) + 4 x^2 (8)
4 x^3 + 32 x^2
REMEMBER to add the exponents if the bases are the same.
4 x^2 ( x ) = 4 x 2 + 1^ = 4 x^3
Also, REMEMBER that the sign we get when we multiply gives us the sign between the terms.
EXAMPLE: โ y (โ 3 y^2 โ 2 y + 6)
Use the Distributive Property to multiply each term inside the parentheses by โ y. REMEMBER that the sign in front of the term goes with the term.
โ y (โ 3 y^2 โ 2 y + 6)
โ y (โ 3 y^2 ) โ y (โ 2 y ) โ y (6)
3 y^3 + 2 y^2 โ 6 y
REMEMBER that we cannot combine terms unless the variable parts are identical. This problem is simplified as far as possible.
EXAMPLE: ab (2 a^2 โ 4 ab โ 6 b^2 )
ab (2 a^2 ) + ab (โ 4 ab ) + ab (โ 6 b^2 )
2 a^3 b โ 4 a^2 b^2 โ 6 ab^3
Donโt forget the rules for exponents!
This instructional aid was prepared by the Tallahassee Community College Learning Commons.
Multiplying a Polynomial by a Polynomial
Multiplication of polynomials can be accomplished by using a horizontal format and the Distributive Property, or by using a vertical format. We will use the vertical formatโthe process is similar to multiplication of real numbers.
EXAMPLE: ( y^2 โ 2 y + 7)( y โ 2)
REWRITE in vertical format. (^2 2 ) 2
y y y
Multiply each term by โ2: 2
2
y y y y y
Now multiply each term by y. Be sure to keep like terms lined up.
2
2 3 2 3 2
y (^) y y y y y y y y y y
If you compare multiplication of polynomials to long multiplication of integers you will see that the steps are very similar.
ร
2
2
x x x x x
ร
2
2 3 2
x x x x x x x x
ร
2
2 3 2 3 2
x x x x x x x x x x x
Now combine like terms
Multiply by 3 Multiply by โ 2
Multiply by 2, keeping place values aligned
Multiply by 4 x , keeping like terms together
Now add to get the total.
Add by combining like terms
c. (^) โ x (โ 2 x^4 โ 3 x^2 + 2) h. (^) (โ 2 x^2 + 3 x + 8)( x + 7)
d. (^) (3 y^3 โ 2 y^2 + 5)4 y i. (^) (4 a^3 โ 2 a + 5)( a + 6)
e. (^) โ 3 ab ( a^2 โ 5 ab + 3 b^2 ) j. (^) (5 y^3 + 2 y^2 โ 9)(2 y โ 3)
KEY:
a. โ 6 a^4 โ 12 a^3 b. 6 y^3 โ 8 y c. 2 x^5 + 3 x^3 โ 2 x d. 12 y^4 โ 8 y^3 + 20 y e. โ 3 a^3 b + 15 a^2 b^2 โ 9 ab^3 f. โ 2 x^3 + 7 x^2 โ 7 x + 2 g. 15 y^3 โ 2 y^2 โ 18 y + 8 h. โ 2 x^3 โ 11 x^2 + 29 x + 56 i. 4 a^4 + 24 a^3 โ 2 a^2 โ 7 a + 30 j. 10 y^4 โ 11 y^3 โ 6 y^2 โ 18 y + 27
