ADDING, SUBTRACTING, & MULTIPLYING POLYNOMIALS

ERROR ANALYSIS

A student simplified 6a³ -a³ and got a 6 as a result. Write an explanation of the student’s error using the words coefficient and like terms.

CHALLANGE

A 35-centimeter stick is cut into three pieces so that the two end pieces are each equal in length to one-third of the middle piece. How long is each piece?

MULTIPLYING POLYNOMIALS

3a(6b + 7) (3a)(6b) + (3a)(7) 18ab + 21a

2x(3x² + x – 4) (2x)(3x²) + (2x)(x) + (2x)(-4) 6x³ + 2x² + -8x

YOUR TURN

-4a²(-2a² + 3ab – 2b + 5)

MULTIPLYING BINOMIALS

(x + 2)(x + 3) x(x+ 3) + 2(x + 3) (x)( x)+ (x)(3) + 2(x) + 2(3) x² + 3x + 2x + 6 x² + 5x + 6

FOIL METHOD

(x + 3)(x + 2) + + +

x² + 2x + 3x + 6 x² + 5x+ 6

first2)

outer(xinside

last

First

x·xOuter

x ·2

Inside

3 · x

Last

3 · 2

YOUR TURN – USE FOIL METHOD

1. (x + 4)(x - 3) 2. (2x + 2)(x + 6)

3. (3ab² + b)(-2ab² - b) 4. (4xy + 3)(2x²y + 1)

FOIL

1. (x + 4)(x – 4) 2. (x + 3)(x – 3)

3. (3y – 1)(3y + 1) 4. (2x² + 3)(2x² - 3)

5. (2x +y) (2x -y) 6. (x²y - 2)(x²y + 2)

FOIL

1.(y + 1)(y + 1) 2. (x + 2)²

3. (2x - 3)² 4. .(y² + 1)²

5.(2x – y)² 6. .(2y² - 2x)²

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

MULTIPLYING POLYNOMIALS : HORIZONTALLY

(3x2 – 2x – 4)(x – 3) = (3x2)(x – 3) + (-2x)(x – 3) + (-4)(x –

3) = (3x3 – 9x2) + (-2x2 + 6x) + (-4x + 12) = 3x3 – 9x2 – 2x2 + 6x – 4x +12 = 3x3 – 11x2 + 2x + 12

MULTIPLYING POLYNOMIALS: VERTICALLY

(-x2 + 2x + 4)(x – 3)= -x2 + 2x + 4

* x – 3 3x2– 6x – 12

-x3 + 2x2 + 4x_____ -x3 + 5x2 – 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

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

x3 + 15x2 + 75x + 125