A.3 Polynomials and Factoring In the following polynomial, what is the degree and leading coefficient? 4x 2 - 5x 7 - 2 + 3x Degree = Leading coef. = 7.

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A.3 Polynomials and Factoring In the following polynomial, what is the degree and leading coefficient? 4x 2 - 5x x Degree = Leading coef. = 7 -5 Ex. 1 Adding polynomials (7x 4 - x 2 - 4x + 2) - (3x 4 - 4x 2 + 3x)First, dist. the neg. = 4x 4 + 3x 2 - 7x + 2

Ex. 2Foil (3x - 2)(5x + 7)= 15x x - 14 Ex. 3The product of Two Trinomials (x + y - 2)(x + y + 2) =x 2 + xy + 2x + xy + y 2 + 2y -2x - 2y - 4 = x 2 + 2xy + y 2 - 4

Pascal’s Triangle Pascal’s Triangle can be used to expand polynomials that look like.... (a + b) 0 (a + b) 1 (a + b) 2 (a + b) 3 (a + b) 4 (a + b) 5 (a + b) 6

Ex.4Expand (x + y) 3 The row that matches up with this example is row 4. It is These are the coef. in front of each term. 1x 3 y 0 + 3x 2 y 1 + 3x 1 y 2 + 1x 0 y Notice that the sum of the exponents always add up to three.

Let’s do (a + b) 5 What line of coef. are we going to use? a 5 + 5a 4 b + 10a 3 b a 2 b 3 + 5ab 4 + b 5

One more… (2x - 3y) 4 Write down the coef. first a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4 Now let a = 2x and b = -3y (2x) 4 + 4(2x) 3 (-3y) + 6(2x) 2 (-3y) 2 + 4(2x)(-3y) 3 + (-3y) 4 16x x 3 y + 216x 2 y xy y 4 Day 1

Removing Common Factors Ex.5 6x 3 - 4x =2x(3x 2 - 2) (x - 2)(2x) + (x - 2)(3) =(x - 2)(2x + 3) x 2 = 3(1 - 4x 2 ) = 3(1 - 2x)(1 + 2x)

Factoring the Difference of Two Squares Ex. 6 (x + 2) 2 - y 2 = 16x = (x y)(x y) or (x - y + 2)(x + y + 2) (4x 2 - 9)(4x 2 + 9) (2x + 3)(2x - 3)(4x 2 + 9) Factoring Perfect Trinomials Ex.716x 2 + 8x + 1 = (4x + 1)(4x + 1) = (4x + 1) 2

Ex. 8 Factorx 2 - 7x + 12 (x - 3)(x - 4) 2x 2 + x - 15 (2x - 5)(x + 3) Factoring the Sum and Difference of Cubes

Ex.9x = (x) 3 - (3) 3 Let u = x and v = 3 Plug these into the diff. of cubes equation

Ex. 10Factor 3x First, factor out a 3. 3(x ) Next, write each term as something cubed and set them equal to a and b. 3((x) 3 + (4) 3 )Let a = x and b = 4

Ex.11Factoring by Grouping x 3 - 2x 2 - 3x + 6 { { What can we factor out of the first two terms? And the second two terms? x 2 (x - 2) - 3(x - 2)Did you remember to factor a negative from the +6? Now what does each group have in common? Now factor it out. (x - 2)(x 2 - 3)