1. Multiplication of Polynomials
Chapter 7.3

2. Multiplying a Polynomial by a Monomial (Objective. 1)
Simply use the “Distributive Property” e.g.
2x (x+3) =
2x(x) + 2x(3)=
2x x

3. More examples. (-2y+3)(-4y) = (-4y)(-2y) + (-4y)(3) = 8y2 -12y
a2(3a2+2a-7)= a2(3a2)+ a2(2a) + a2(-7) =
3a4+2a3-7a2
2x(4x3 +5x2 +2x -9) =
2x(4x3) + 2x(5x2) + 2x(2x) + 2x(-9)=
8x4+10x3+4x2-18x

4. Multiply two Polynomials (Objective. 2)
Use repeated distributive Property.
(x-2)(4x3+5x2+2x-9) = 1st Distribute x
x(4x3)+x(5x2)+x(2x)+x(-9) = 2nd Distribute -2
(-2)(4x3)+(-2)(5x2)+(-2)(2x)+(-2)(-9)
4x4 +5x3 +2x2 -9x x3 -10x2 -4x +18
Combine like Terms =
4x4-3x3-8x2-13x+18

5. Vertical Method 4x3 + 5x2 + 2x – 9 x – 2 -8x3 -10x2 -4x+18

6. More Examples 2y3 + 2y2 -3 3y – 1 Changes the sign: -2y3 -2y2 +3
6y4 +4y3 -2y2 -9y +3

7. (3x3 -2x2 +x -3)(2x+5) 3x3 -2x2 + x -3 2x + 5 15x3 -10x2 +5x - 15

8. You Try 1. x(3x3 -2x2 +x -3) 2. -3x3(3x3 -2x2 +4x -5)

9. Answers 1. 3x4 -2x3 + x2 -3x 2. -9x6 + 6x5 -12x4 +15x3

10. Multiply Two Binomials Objective 3
Multiply (2x+3)(x+5) using vertical method.
2x+3
x + 5
10x +15
2x2 + 3x +0
2x2 +13x +15

11. Multiply Two Binomials Using Distributive Property “FOIL”
F First term times First Term
O Outer term times Outer term.
I Inner term times Inner term.
L Last term times Last term.

12. F(2x)(x) +O (2x)(5) +I (3)(x) +L (3)(5) =
Expand (2x+3)(x+5)
F(2x)(x) +O (2x)(5) +I (3)(x) +L (3)(5) =
2x x x =
2x x

13. EXAMPLES (4x-3)(3x-2) = (4x)(3x)+ (4x)(-2)+ (-3)(3x)+ (-3)(-2) =

14. Expand (3a+2b)(3a-5b) (3a)(3a)+ (3a)(-5b) + (2b)(3a)+ (2b)(-5b) =
9a ab ab b2 =
9a ab b2 =
9a ab -10b2

15. Expand (6a+b)(3a-9b) (6a)(3a)+ (6a)(-9b)+ (b)(3a) + (b)(-9b) =
18a ab ab b2 =
18a ab b2 =
18a ab -9b2

16. Multiply Binomials That Have Special Products
There are a couple of procedures that involve FOIL that do not require all the steps or that follow a pattern.
The first is the Sum and the Difference of Two Terms.
If the binomials to be expanded are identical except that the signs are opposite, then the middle term subtracts out.

17. Expand (a+b)(a-b) Using the FOIL method.
(a)(a) + (a)(b) + (a)(-b) (b)(-b) =
a ab ab b2 =
a b =
The middle term subtracts out.
a b2

18. Examples (2a+5c)(2a-5c)
(2a) (5c)2 = You only have to square first and last terms, and the sign is negative.
4a c2
Expand (x+1)(x-1)
(x)2 - (1)2 = x2 - 1

19. The square of a Binomial
FOIL (a+b)2 = (a+b)(a+b)
(a)(a)+ (a)(b) + (b)(a) (b)(b) =
a ab ab b2 =
a ab b2 =
a ab + b2
Note pattern:
(1st term)2 +2(first* last term) + (last term)2

20. FOIL (a-b)2 = (a-b)(a-b)
(a)(a)+ (a)(-b) + (-b)(a) (-b)(-b) =
a ab ab b2 =
a ab b2 =
a ab + b2
Note pattern:
(1st term)2 -2(first* last term) + (last term)2

21. Examples Expand (4x+5d)2 Recall Pattern. ( )2 + 2( ) + ( )2 =
( ) ( ) ( )2 =
(4x) (4x*5d) (5d)2
16x xd + 25d2

22. Expand (3x+2y)2 Recall Pattern. ( )2 + 2( ) ( )2 =
( ) ( ) ( )2 =
(3x) (3x*2y) (2y)2
9x xy y2

23. Expand (6x-y)2 Recall Pattern. ( )2 - 2( ) + ( )2 =
( ) ( ) ( )2 =
(6x) (6x*y) (-y)2
36x xy + y2

24. Now YOU TRY 1. (x-3)(x+3) 1. x2 - 9 2. (2x+4y)(2x-4y) 2. 4x2 - 16y2

25. Now YOU TRY 4. (9x+5z)2 4. 81x2 + 90xz + 25z2 5. (4x-5d)2
5. 16x xd + 25d2
6. (11b-12c)2
6. 121b bc c2