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Mastering Polynomial Factoring: Steps and Patterns Explained

This guide provides a comprehensive overview of factoring polynomials, suitable for high school students. Follow detailed steps, beginning with identifying the greatest common factor (GCF) and determining if the polynomial is a binomial or trinomial. Explore specific patterns for binomials, including the difference and sum of perfect squares and cubes. Learn effective grouping techniques for polynomials with more than three terms. Practice trial and error with trinomial factors, and conclude with verifying your results. Get ready to confidently factor any polynomial!

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Mastering Polynomial Factoring: Steps and Patterns Explained

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  1. Factoring Polynomials Miss Ankrom Glenvar High School

  2. Study the polynomial carefully!

  3. Do the terms of the polynomial shareacommon factor?

  4. Is the coefficient of the term of highest degree a negative number?

  5. NowLet’s GO!!

  6. Step # 1 Factor out the Greatest Common Factor Or Factor out -1. Then rewrite it as the product of the GCF and a polynomial.

  7. Again studythe polynomial carefully!

  8. Decisions must now be made!!

  9. The number of terms Matters!! Is the polynomial a Binomial or a Trinomial?

  10. Step # 2Binomial Patterns There are three possible factor patterns for binomials. • The difference of two perfect squares • The sum of two perfect cubes • The difference of two perfect cubes

  11. Factor Patterns These are “cookie cutter” patterns • If the polynomial “fits” the pattern, it factors! • If it doesn’ t “fit”, it will not factor!

  12. The Difference of Two Perfect Squares A2 – B2 (a + b)(a – b)

  13. The sum of two perfect cubes A3 + B3 (a + b)(a2 – ab + b2)

  14. The difference of two perfect cubes A3 – B3 (a + b)(a2 – ab + b2)

  15. Step # 3 Is the polynomial a trinomial? ax2 + bx + c

  16. Trial and/or Error • Try all possible combinations of the two binomial factors until you find the right combination. • Ifno combination multiplies back to the original polynomial, then the polynomial does not factor.

  17. Doesthe polynomial have more than three terms?

  18. Step # 4Grouping There are two types of grouping with four terms. • Two by Two grouping (“Noah’s Ark”) • Three by One grouping

  19. Summary • Repeat each step on the factors you find until the polynomial is factored completely. • Rememberthat the sum of the degrees of the factors must equal the degree of the original polynomial. • Thefactor set is the set of integers. • You can always check your answer by multiplying the factors together to show that the product is the originalpolynomial.

  20. Step #1: Factor out the GCF or -1 Step #2: If it is a binomial, check the three factor patterns below. Difference of two squares Sum of two cubes Difference of two cubes Step #3: If it is a trinomial, try to factor it into the product of 2 binomial factors by using trial and/or error. Step #4: If there are more than three terms, try grouping the terms and then using the previous factoring rules. Factoring Polynomials Completely

  21. Congratulations! You can now successfully factor any polynomial. HAVE FUN!

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