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Section 5.6

Section 5.6. Summary of Factoring. Objectives. Guidelines for Factoring Polynomials Factoring Polynomials. Example. Factor 6 x 3 – 36 x 2 . Solution Step 1: The greatest common factor is 6 x 2 . 6 x 3 – 36 x 2 = 6 x 2 ( x – 6 )

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Section 5.6

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  1. Section 5.6 • Summary of Factoring

  2. Objectives Guidelines for Factoring Polynomials Factoring Polynomials

  3. Example Factor 6x3 – 36x2. Solution Step 1: The greatest common factor is 6x2. 6x3 – 36x2 = 6x2(x – 6) Step 2: The binomial, x – 6, cannot be factored further. Step 3: The completely factored polynomial is 6x2(x – 6).

  4. Example Factor 3x3 – 75x. Solution Step 1: The greatest common factor is 3x. 3x3 – 75x= 3x(x2 – 25) Step 2: We can factor the binomial x2 – 25 as a difference of squares. Step 3: The completely factored polynomial is 3x(x – 5)(x + 5) 3x(x2 – 25) = 3x(x – 5)(x + 5)

  5. Example Factor 2x4 – 162. Solution Step 1: The greatest common factor is 2. 2x4 – 162= 2(x4 – 81) Step 2: We can factor x4 – 81 as the difference of squares, twice. The sum of squares, x2 + 9, cannot be factored further. Step 3: The completely factored polynomial is 2(x – 3)(x + 3)(x2 + 9). 2(x4 – 81) = 2(x2 – 9)(x2 + 9) = 2(x – 3)(x + 3)(x2 + 9)

  6. Example Factor 8x3 + 40x2 + 50x. Solution Step 1: The greatest common factor is 2x. 8x3 + 40x2 + 50x= 2x(4x2 + 20x + 25) Step 2: The trinomial 4x2 + 20x + 25 is a perfect square trinomial. Step 3: The completely factored polynomial is 2x(2x + 5)2 2x(4x2 + 20x + 25) = 2x(2x + 5)(2x + 5)

  7. Example Factor −2y4 − 128y. Solution Step 1: The greatest common factor is −2y. −2y4 − 128y= −2y(y3 + 64) Step 2: The binomial y3 + 64 can be factored as a sum of cubes. Step 3: The completely factored polynomial is −2y(y + 4)(y2 – 4y + 16). −2y(y3 + 64) = −2y(y + 4)(y2 – 4y + 16)

  8. Example Factor x3 − 7x2 – 4x + 28. Solution Step 1: There are no common factors. Step 2: Because there are four terms, we apply grouping. Step 3: The completely factored polynomial is (x − 2)(x + 2)(x − 7) x3 − 7x2 – 4x + 28 = (x3 − 7x2) – (4x − 28) = x2(x − 7) – 4(x − 7) = (x2 − 4)(x − 7) difference of two squares

  9. Example Factor 12x2 − 75y2. Solution Step 1: The greatest common factor is 3. 12x2 − 75y2 = 3(4x2 − 25y2) Step 2: We can factor the binomial as the difference of squares. Step 3: The completely factored polynomial is 3(2x − 5y)(2x + 5y) 3(4x2 − 25y2) = 3(2x − 5y)(2x + 5y)

  10. Example Factor 3x5 − 27x3 – 3x2 + 27. Solution Step 1: The greatest common factor is 3. 3x5 − 27x3 – 3x2 + 27= 3(x5 − 9x3 − x2 + 9) Step 2: Factor by grouping. 3(x5 − 9x3 − x2 + 9) = 3(x3(x2 – 9) − 1(x2 – 9)) = 3(x3 − 1)(x2 – 9)

  11. Example (cont) Factor 3x5 − 27x3 – 3x2 + 27. Step 2: Factor as the difference of two cubes and the difference of two squares. The complete factorization is: 3(x − 1)(x2 + x + 1)(x – 3)(x + 3) 3(x3 − 1)(x2 – 9) = 3(x − 1)(x2 + x + 1)(x – 3)(x + 3)

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