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A.3 Polynomials and Factoring

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 -5. Ex. 1 Adding polynomials. (7x 4 - x 2 - 4x + 2) - (3x 4 - 4x 2 + 3x). First, dist. the neg.

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A.3 Polynomials and Factoring

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  1. A.3 Polynomials and Factoring In the following polynomial, what is the degree and leading coefficient? 4x2 - 5x7 - 2 + 3x Degree = Leading coef. = 7 -5 Ex. 1 Adding polynomials (7x4 - x2 - 4x + 2) - (3x4 - 4x2 + 3x) First, dist. the neg. = 4x4 + 3x2 - 7x + 2

  2. Ex. 2 Foil (3x - 2)(5x + 7) = 15x2 + 11x - 14 Ex. 3 The product of Two Trinomials (x + y - 2)(x + y + 2) = x2 + xy + 2x + xy + y2 + 2y -2x - 2y - 4 = x2 + 2xy + y2 - 4

  3. Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 (a + b)0 (a + b)1 (a + b)2 (a + b)3 (a + b)4 (a + b)5 (a + b)6 Pascal’s Triangle can be used to expand polynomials that look like....

  4. Ex. 4 Expand (x + y)3 The row that matches up with this example is row 4. It is 1 3 3 1 These are the coef. in front of each term. 1 3 3 1 1x3y0 + 3x2y1 + 3x1y2 + 1x0y3 Notice that the sum of the exponents always add up to three.

  5. Let’s do (a + b)5 What line of coef. are we going to use? 1 5 10 10 5 1 a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5

  6. One more… (2x - 3y)4 Write down the coef. first. 1 4 6 4 1 a4 + 4a3b + 6a2b2 + 4ab3 + b4 Now let a = 2x and b = -3y (2x)4 + 4(2x)3(-3y) + 6(2x)2(-3y)2 + 4(2x)(-3y)3 + (-3y)4 16x4 - 96x3y + 216x2y2 - 216xy3 + 81y4 Day 1

  7. Removing Common Factors Ex. 5 6x3 - 4x = 2x(3x2 - 2) (x - 2)(2x) + (x - 2)(3) = (x - 2)(2x + 3) 3 - 12x2 = 3(1 - 4x2) = 3(1 - 2x)(1 + 2x)

  8. Factoring the Difference of Two Squares Ex. 6 (x + 2)2 - y2 = 16x4 - 81 = (x + 2 - y)(x + 2 + y) or (x - y + 2)(x + y + 2) (4x2 - 9)(4x2 + 9) (2x + 3)(2x - 3)(4x2 + 9) Factoring Perfect Trinomials (4x + 1)(4x + 1) Ex. 7 16x2 + 8x + 1 = = (4x + 1)2

  9. Ex. 8 Factor x2 - 7x + 12 2x2 + x - 15 (x - 3)(x - 4) (2x - 5)(x + 3) Factoring the Sum and Difference of Cubes

  10. Ex. 9 x3 - 27 = (x)3 - (3)3 Let u = x and v = 3 Plug these into the diff. of cubes equation

  11. Ex. 10 Factor 3x3 + 192 First, factor out a 3. 3(x3 + 64) 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

  12. Ex. 11 Factoring by Grouping x3 - 2x2 - 3x + 6 What can we factor out of the first two terms? And the second two terms? { { x2(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)(x2 - 3)

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