Multiplication of Polynomials

Chapter 7.3

Multiplying a Polynomial by a Monomial (Objective. 1)

Simply use the “Distributive Property” e.g.

2x (x+3) = 2x(x) + 2x(3)= 2x2 + 6x

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

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 + -8x3 -10x2 -4x +18 Combine like Terms = 4x4-3x3-8x2-13x+18

Vertical Method 4x3 + 5x2 + 2x – 9

x – 2

-8x3 -10x2 -4x+18 4x4 +5x3 +2x2 -9x _ 4x4 -3x3 -8x2 -13x+18

More Examples 2y3 + 2y2 -3

3y – 1

Changes the sign: -2y3 -2y2 +3 6y4 + 6y3 -9y +0

6y4 +4y3 -2y2 -9y +3

(3x3 -2x2 +x -3)(2x+5) 3x3 -2x2 + x -3

2x + 5

15x3 -10x2 +5x - 15 6x4 -4x3 + 2x2 -6x +0 6x4+11x3 - 8x2 - x - 15

You Try 1. x(3x3 -2x2 +x -3) 2. -3x3(3x3 -2x2 +4x -5) 3. (7x3 +11x2 -6x -13)(2x2 -3) 4. (9x -3)(x3 -3x2 +2x -1) 5. (2x3 -12x2 +3x)(2x2-3) 6. (-x+4)(8x3 +3x2 -13)

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

3. 14x5 +22x4 -33x3 -59x2 +18x +39 4. 9x4 -30x3 +27x2 -15x +3 5. 4x5 -24x4+36x2 -9x

6. -8x4+29x3 +12x2 +13x -52

Multiply Two BinomialsObjective 3

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

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.

Expand (2x+3)(x+5)

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

2x2 + 10x + 3x + 15 =

2x2 + 13x + 15

EXAMPLES (4x-3)(3x-2) = (4x)(3x)+ (4x)(-2)+ (-3)(3x)+ (-3)(-2) = 12x2 + -8x + -9x + 6 = 12x2 - 17x + 6

Expand (3a+2b)(3a-5b) (3a)(3a)+ (3a)(-5b) + (2b)(3a)+ (2b)(-5b) = 9a2 + -15ab + 6ab + -10b2 = 9a2 - 9ab -10b2 =

9a2 -9ab -10b2

Expand (6a+b)(3a-9b) (6a)(3a)+ (6a)(-9b)+ (b)(3a) + (b)(-9b) = 18a2 + -54ab + 3ab + -9b2 = 18a2 - 51ab -9b2 =

18a2 -51ab -9b2

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.

Expand (a+b)(a-b) Using the FOIL method. (a)(a) + (a)(b) + (a)(-b) + (b)(-b) = a2 + ab -ab - b2 = a2 - b2 = The middle term subtracts out. a2 - b2

Examples (2a+5c)(2a-5c) (2a)2 - (5c)2 = You only have to square first

and last terms, and the sign is negative. 4a2 - 25c2

Expand (x+1)(x-1) (x)2 - (1)2 = x2 - 1

The square of a Binomial FOIL (a+b)2 = (a+b)(a+b) (a)(a)+ (a)(b) + (b)(a) + (b)(b) = a2 + ab + ab + b2 = a2 + 2ab + b2 = a2 +2ab + b2

Note pattern: (1st term)2 +2(first* last term) + (last term)2

FOIL (a-b)2 = (a-b)(a-b) (a)(a)+ (a)(-b) + (-b)(a) + (-b)(-b) = a2 - ab - ab + b2 = a2 - 2ab + b2 = a2 - 2ab + b2

Note pattern: (1st term)2 -2(first* last term) + (last term)2

Examples Expand (4x+5d)2 Recall Pattern. ( )2 + 2( ) + ( )2 = (4x)2 + 2(4x*5d) + (5d)2 16x2 + 40xd + 25d2

Expand (3x+2y)2

Recall Pattern. ( )2 + 2( ) ( )2 = (3x)2 + 2 (3x*2y) + (2y)2 9x2 + 12xy + 4y2

Expand (6x-y)2

Recall Pattern. ( )2 - 2( ) + ( )2 = (6x)2 - 2(6x*y) + (-y)2 36x2 - 12xy + y2

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

3. (7x+3)2

3. 49x2 + 42x + 9

Now YOU TRY

4. (9x+5z)2

4. 81x2 + 90xz + 25z2

5. (4x-5d)2

5. 16x2 - 40xd + 25d2

6. (11b-12c)2

6. 121b2 - 264bc + 144c2