5.3 Multiplying Polynomials
Multiplying a Polynomial by a Monomial
To multiply a polynomial by a monomial, use the distributive property to multiply each term in the
polynomial by the monomial.
a ( b + c + d) = a b + ac + ad
Examples
Multiply.
[08]
4y5( -3y3 - 5y2 + 3)
4y5(-3y3) + 4y5(-5y2) + 4y5(3)
-12y8
– 20y7
+ 12y5
2
3
2 2
[20]
-5a b(3ab – 2a b + 3ab5)
(-5a2b)(3ab3) + (-5a2b)(-2a2b2) + (-5a2b)(3ab5)
-15a3b4
+10a4b3
– 15a3b6
.
Note:
When multiplying multivariable terms, it is helpful to multiply the coefficients first, then the
variables in alphabetic order.
Multiplying Polynomials
A method to multiply two polynomials is
1. Multiply each term in the second polynomial by each term in the first polynomial.
2. Combine like terms.
FOIL
When you are multiplying binomials, an alternate view is called FOIL.
Sum the product of the First terms, then the product of the Outer terms, then product of the Inner Terms, and
finally product of the Last terms.
(a + b)(c + d)
first
outer inner
last
a●c + a●d + b●c + b●d
Examples
[32]
( 3a + 4 )( 7a + 2 )
3a●7a + 3a●2 + 4●7a + 4●2
21a2
+ 6a + 28a + 8
21a2
+ 34a
+ 8
[42]
( 3x2 + 4y2 )( x2 – 5y2 )
3x2 ● x2 + 3x2 ●(–5y2) + 4y2 ● x2 + 4y2 ●(–5y2)
3x4 + (-15x2y2) + 4x2y2 + (-20y4)
3x4
- 11x2y2
- 20y4
[44]
( 3x – 1 ) ( 2x2 – 2x + 1 )
2x2 – 2x + 1
×
3x - 1
-2x2 + 2x - 1
6x3 - 6x2 + 3x
.
6x3 - 8x2 + 5x - 1
Special Cases
Squaring Binomials
Multiplying Conjugates
conjugates – binomials that are the sum
and difference of the same terms
Examples:
3x – 2 and 3x + 2
-4y + 3 and -4y - 3
Squaring a Binomial
If a and b are real numbers, variables, or expressions, then
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
Examples:
[58]
( 3k + 2 )2 = (3k)2 + 2(3k)(2) + (2)2
= 9k2 + 12k + 4
2 2
[62]
( 6 - 7n ) = (6)2 - 2(6)(7n2) + (7n2)2
= 36 - 84n2 + 49n4
Multiplying Conjugates
If a and b are real numbers, variables, or expressions, then
Examples
[78]
( 3p + 4 )( 3p – 4 ) = ( 3p )2 - ( 4 )2
= 9p2 - 16
[82]
{ (y+3) + z}{(y+3) - z} = (y+3)2 – (z)2
= (y)2+2(y)(3)+(3)2–z2
= y2 + 6y + 9 – z2
(a + b)(a – b) = a2 – b2.