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Factoring Polynomials

Factoring Polynomials. The Greatest Common Factor. Factors. Factors (either numbers or polynomials) When an integer is written as a product of integers, each of the integers in the product is a factor of the original number.

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Factoring Polynomials

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  1. Factoring Polynomials

  2. The Greatest Common Factor

  3. Factors Factors (either numbers or polynomials) When an integer is written as a product of integers, each of the integers in the product is a factor of the original number. When a polynomial is written as a product of polynomials, each of the polynomials in the product is a factor of the original polynomial. Factoring – writing a polynomial as a product of polynomials.

  4. Greatest Common Factor Greatest common factor – largest quantity that is a factor of all the integers or polynomials involved. Finding the GCF of a List of Integers or Terms • Prime factor the numbers. • Identify common prime factors. • Take the product of all common prime factors. • If there are no common prime factors, GCF is 1.

  5. Greatest Common Factor Example Find the GCF of each list of terms. • x3 and x7 x3 = x ·x·x x7 = x ·x·x·x ·x·x·x So the GCF is x · x· x = x3 • 6x5 and 4x3 6x5 = 2 · 3 · x · x· x 4x3 = 2 · 2 ·x ·x·x So the GCF is 2·x ·x·x = 2x3

  6. Greatest Common Factor Example Find the GCF of the following list of terms. a3b2, a2b5 and a4b7 a3b2 = a ·a·a· b· b a2b5 = a · a· b· b · b· b· b a4b7 = a · a· a· a· b· b · b· b· b· b· b So the GCF is a · a· b· b = a2b2 Notice that the GCF of terms containing variables will use the smallest exponent found amongst the individual terms for each variable.

  7. Factoring out the GCF Example 1) 6x3 – 9x2 + 12x = 3· x· 2 ·x2 – 3·x· 3 ·x + 3·x· 4 = 3x(2x2 – 3x + 4)

  8. Factoring out the GCF Example 2) 14x3y + 7x2y – 7xy = 7 ·x·y· 2 ·x2 + 7·x·y· x – 7·x·y· 1 = 7xy(2x2 + x – 1)

  9. Factoring Trinomials of the Form x2 + bx + c by Grouping

  10. Factoring by Grouping Factoring a Four-Term Polynomial by Grouping • Arrange the terms so that the first two terms have a common factor and the last two terms have a common factor. • For each pair of terms, use the distributive property to factor out the pair’s greatest common factor. • If there is now a common binomial factor, factor it out. • If there is no common binomial factor in step 3, begin again, rearranging the terms differently. • If no rearrangement leads to a common binomial factor, the polynomial cannot be factored. • Remember: all have a common factor of 1!

  11. Factoring by Grouping Factoring polynomials often involves additional techniques after initially factoring out the GCF. One technique is factoring by grouping. Example Factor xy + y + 2x + 2 by grouping. Notice that, although 1 is the GCF for all four terms of the polynomial, the first 2 terms have a GCF of y and the last 2 terms have a GCF of 2. xy + y + 2x + 2 y(x + 1)+ 2(x + 1)= (x + 1)(y + 2)

  12. Factoring by Grouping Example Factor the following polynomial by grouping. • x3 + 4x + x2 + 4 = x ·x2 + x· 4 + 1· x2 + 1· 4 = x(x2 + 4) + 1(x2 + 4) = (x2 + 4)(x + 1)

  13. Factoring by Grouping Example Factor the following polynomial by grouping. • 2x3 – x2 – 10x + 5 = x2 · 2x – x2· 1 – 5· 2x – 5· (– 1) = x2(2x – 1) – 5(2x – 1) = (2x – 1)(x2 – 5)

  14. Factoring by Grouping Example Factor 2x – 9y + 18 – xy by grouping. Neither pair has a common factor (other than 1). So, rearrange the order of the factors. 2x + 18 – 9y – xy = 2 · x + 2· 9 – 9 · y – x· y = 2(x + 9) – y(9 + x) = 2(x + 9) – y(x + 9) = (make sure the factors are identical) (x + 9)(2 – y)

  15. Lab • Textbook page:

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