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Understanding the Greatest Common Factor (GCF) of Polynomials: A Comprehensive Guide

This guide delves into the concept of the Greatest Common Factor (GCF) of polynomials, highlighting its significance and applications. We will explore how to find the GCF with variables, emphasizing that it corresponds to the smallest exponent of a common variable. Examples will illustrate the process, including polynomial expressions like 12x³, 16x², and 45a⁵. The method of factoring out the GCF will be detailed, guiding you on how to factor polynomials using grouping techniques and ensuring each term is divided correctly.

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Understanding the Greatest Common Factor (GCF) of Polynomials: A Comprehensive Guide

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  1. Polynomials

  2. GCF Greatest Common Factor

  3. What is a GCF of a polynomial?

  4. GCF with Variables Note: With variables, the GCF will always be the smallest exponent of a common variable Examples: 12x3, 16x2 45a5, 50a7 GCF = 4x2 GCF = 5a5

  5. Factor out the GCF Put the GCF outside of (parenthesis). Divide each term by the GCF. You will always have the same numbers of terms you start with

  6. 16x2 – 8x 8x(2x – 1)

  7. 10x – 10y 10(x – y)

  8. 8r2 – 24r 8r(r – 3)

  9. 6n2 + 15n 3n(2n + 5)

  10. 6x3 – 9x2 + 3x 3x(2x2 – 3x + 1)

  11. 2a3 – 6a 2a(a2 – 3)

  12. 8y3 – 20y2 + 12y 4y(2y2 – 5y + 3)

  13. 7x3 – 28x2 7x2(x – 4)

  14. 4m3 – 20m 4m(m2 – 5)

  15. 3x(x + 2) – 2(x + 2) (x + 2)(3x – 2)

  16. 5z(z – 6) + 4(z – 6) (z – 6)(5z + 4)

  17. Grouping

  18. 1. Group the 1st two terms and the 2nd two terms Steps to Factor by Grouping 4 terms 2. Factor out the GCF of each group 3. Write down the common parenthesis 4. In another parenthesis, write the GCFs 5. Check to see if the parenthesis can factor again

  19. x3 + 12x2 – 3x – 36 x2(x + 12) – 3(x + 12) (x + 12)(x2 – 3)

  20. y3 – 14y2 + y – 14  (y3 – 14y2) + (y – 14) y2(y – 14) + 1(y – 14) (y – 14)(y2 + 1)

  21. m3 – 6m2 + 2m – 12 (m3 – 6m2) + (2m – 12) m2(m – 6) + 2(m – 6) (m – 6)(m2 + 2)

  22. p3 + 9p2 + 4p + 36 (p3 + 9p2) + (4p + 36) p2(p + 9) + 4(p + 9) (p + 9)(p2 + 4)

  23. x3 + x2 + 5x + 5 (x3 + x2) + (5x + 5) x2(x + 1) + 5(x + 1) (x + 1)(x2 + 5)

  24. x3 – 3x2 – 5x + 15  (x3 – 3x2) + (-5x + 15) x2(x – 3) – 5(x – 3) (x – 3)(x2 – 5)

  25. 3x3 – 3x2 + x – 1  (3x3 – 3x2) + (x – 1) 3x2(x – 1) + 1(x – 1) (x – 1)(3x2 + 1)

  26. t2 + 2t + 3kt + 6k (t + 2)(t + 3k)

  27. x2 + 3x + xk + 3k (x + 3)(x + k)

  28. ad + 3a – d2 – 3d (d + 3)(a – d)

  29. 2ab + 14a + b + 7 (b + 7)(2a + 1)

  30. CW/HW - Textbook p. 95 #1 – 18

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