6.3 Adding, Subtracting, & Multiplying Polynomials

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To + or - , + or – the coeff. of like terms!Vertical format :

Add 3x3+2x2-x-7 and x3-10x2+8.

3x3 + 2x2 – x – 7 + x3 – 10x2 + 8 Line up

like terms

4x3 – 8x2 – x + 1

Horizontal format : Combine like terms

(8x3 – 3x2 – 2x + 9) – (2x3 + 6x2 – x + 1)=

(8x3 – 2x3)+(-3x2 – 6x2)+(-2x + x) + (9 – 1)=

6x3 + -9x2 + -x + 8 =

6x3 – 9x2 – x + 8

Examples: Adding & Subtracting

(9x3 – 2x + 1) + (5x2 + 12x -4) =

9x3 + 5x2 – 2x + 12x + 1 – 4 =

9x3 + 5x2 + 10x – 3

(2x2 + 3x) – (3x2 + x – 4)=

2x2 + 3x – 3x2 – x + 4 =

2x2 - 3x2 + 3x – x + 4 =

-x2 + 2x + 4

Multiplying Polynomials: Vertically

(-x2 + 2x + 4)(x – 3)=

-x2 + 2x + 4* x – 3

3x2 – 6x – 12 -x3 + 2x2 + 4x

-x3 + 5x2 – 2x – 12

Multiplying Polynomials : Horizontally

(x – 3)(3x2 – 2x – 4)=

(x – 3)(3x2)

+ (x – 3)(-2x)

+ (x – 3)(-4) =

(3x3 – 9x2) + (-2x2 + 6x) + (-4x + 12) =

3x3 – 9x2 – 2x2 + 6x – 4x +12 =

3x3 – 11x2 + 2x + 12

Multiplying 3 Binomials :

(x – 1)(x + 4)(x + 3) =

FOIL the first two:

(x2 – x +4x – 4)(x + 3) =

(x2 + 3x – 4)(x + 3) =

Then multiply the trinomial by the binomial

(x2 + 3x – 4)(x) + (x2 + 3x – 4)(3) =

(x3 + 3x2 – 4x) + (3x2 + 9x – 12) =

x3 + 6x2 + 5x - 12

Some binomial products appear so much we need to recognize the patterns!

Sum & Difference (S&D):

(a + b)(a – b) = a2 – b2

Example: (x + 3)(x – 3) = x2 – 9

Square of Binomial:

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

(a - b)2 = a2 – 2ab + b2

Last PatternLast Pattern

Cube of a Binomial(a + b)3 = a3 + 3a2b + 3ab2 + b3

(a – b)3 = a3 - 3a2b + 3ab2 – b3

Example:Example:

(x + 5)3 =

a = x and b = 5

x3 + 3(x)2(5) + 3(x)(5)2 + (5)3 =

x3 + 15x2 + 75x + 125