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Chapter R Section 5: Factoring Polynomials

Chapter R Section 5: Factoring Polynomials. In this section, we will… Factor out the GCF (Greatest Common Factor) Factor by Grouping Factor Trinomials of the Form Factor Trinomials of the Form Factor the Sum and Difference of Two Perfect Cubes.

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Chapter R Section 5: Factoring Polynomials

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  1. Chapter R Section 5: Factoring Polynomials • In this section, we will… • Factor out the GCF (Greatest Common Factor) • Factor by Grouping • Factor Trinomials of the Form • Factor Trinomials of the Form • Factor the Sum and Difference of Two Perfect Cubes

  2. In the previous section, we learned how to multiply polynomials; in this section, we will reverse the operation of multiplication by finding the factors of a known product. If one number a divides evenly into another number b, then a is called a factor of b. example: Because 3 divides evenly into 24, 3is a factor of 24. multiply the polynomials factor the polynomial We are, in essence, un-distributing R.5 Factoring Polynomials: Factor Out the GCF

  3. Examples: Factor out the GCF (greatest common factor). R.5 Factoring Polynomials: Factor Out the GCF

  4. Examples: Factor by grouping. R.5 Factoring Polynomials: Factor by Grouping

  5. multiply the polynomials factor the polynomial Factoring Trinomials with a leading coefficient of 1: • Write the trinomial in descending order (for the first variable alphabetically) • Write the factorizations of the third term of the trinomial • Pick the factorization where the sum of the factors is equal to the coefficient of the middle term. sum product We are, in essence, un-FOIL-ing R.5 Factoring Polynomials: Factor Trinomials of the Form x + bx + c 2

  6. Examples: Factor each trinomial. R.5 Factoring Polynomials: Factor Trinomials of the Form x + bx + c 2

  7. Examples: Factor each trinomial R.5 Factoring Polynomials: Factor Trinomials of the Form x + bx + c 2

  8. Factoring Trinomials with a leading coefficient other than 1 (2 methods) Key Number Method • Write the trinomial in descending order • Find the key number: • Find two factors of the key number whose sum is b • Use those factors as the coefficients of two terms to replace the middle term • Factor by grouping Guess-and-Check Method • Write the trinomial in descending order • Write the factorizations that you know • Systematically guess-and-check factorization possibilities where the sum of the factors is equal to the coefficient of the middle term. (Be sure to check!) R.5 Factoring Polynomials: Factor Trinomials of the Formax + bx + c 2

  9. Examples: Factor each trinomial. If a trinomial has the form with integer coefficients and . we can test to see whether it is factorable: If the value of is a perfect square (e.g. 0, 1, 4, 9, 16,…) then the trinomial can be factored using integers. R.5 Factoring Polynomials: Factor Trinomials of the Formax + bx + c 2

  10. Optional: If you know that a trinomial is factorable (because you used the discriminant , but you cannot find the actual factors…the answer can be found by using the graphing utility on your TI and using the zeros. Example: We will find the factors of the trinomial using the TI • Enter the expression to be factored into • Press (the viewing window may need to be adjusted in order to see where the graph crosses the x-axis) • Press [CALC menu] • For each zero, select 2:zero and answer the three questions asked Algebra: Y= GRAPH 2ND TRACE Check to see if the factors are correct! R.5 Factoring Polynomials: Factor Trinomials of the Formax + bx + c 2

  11. ‘sop Examples: Factor the sum or difference of two cubes. R.5 Factoring Polynomials: Factor the Sum and Difference of Two Perfect Cubes

  12. Factor-ific Flow Chart – Cut off this slide and use as a reference. • Write the polynomial in descending order (for the first variable alphabetically) Is the leading coefficient negative? yes Factor out a -1 (or –GCF) Question 2 no Is there a common factor? Question 2 yes Factor out the GCF Question 3 no Determine if it a difference of squares, difference of cubes or sum of cubes? two How many terms are there? Question 3 three Try factoring by trial-and-error (or use key number method) four Try factoring by grouping. Check each of the factors to see if we can factor further. R.5 Factoring Polynomials

  13. Examples: Factor each polynomial completely. R.5 Factoring Polynomials

  14. Examples: Factor each polynomial completely. R.5 Factoring Polynomials

  15. Independent Practice You learn math by doing math. The best way to learn math is to practice, practice, practice. The assigned homework examples provide you with an opportunity to practice. Be sure to complete every assigned problem (or more if you need additional practice). Check your answers to the odd-numbered problems in the back of the text to see whether you have correctly solved each problem; rework all problems that are incorrect. Read pp. 49-55 Homework: pp. 56-57 #9-21 odds, 25, 27, 29, 33, 35, 37, 43-53 odds, 59-71 odds, 79-107 odds Pretend you’re starring in a reality show about a kid who can make his dreams come true if he works hard and gets good grades. R.5 Factoring Polynomials

  16. R.5 Factoring Polynomials

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