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Factoring Polynomials

Factoring Polynomials. A presentation for the greatest Algebra I kids at RJR By Mrs. Sexton. Main Menu. Rules. Step by Step. Easy Problems. Medium Problems. Hard Problems. Word Problems. Division of polynomial by monomial. Find dimensions when area is given.

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Factoring Polynomials

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  1. Factoring Polynomials A presentation for the greatest Algebra I kids at RJR By Mrs. Sexton

  2. Main Menu Rules Step by Step Easy Problems Medium Problems Hard Problems Word Problems Division of polynomial by monomial Find dimensions when area is given

  3. Rules for Factoring Polynomials

  4. Step by Step • Is there a GCF? • Yes • Factor as the product of the GCF and one other factor—i.e. GCF•(the other factor). Look at the other factor and go to the next step below with it. • No • Go the the next step. • Is it a binomial? • Yes • Is it a difference of two squares? (a2-b2) • Yes—Factor as (a+b)(a-b). • No—It can’t be factored any more. • No • Go to the next step.

  5. Is it a trinomial? • Yes • Do you recognize it as a pattern for a perfect square trinomial? (a2+2ab+b2) or (a2-2ab+b2) • Yes—Factor as (a+b)2 or (a-b)2 • No—Go to next step. • Use the ac and b pattern to look for factors. • Can you find factors of ac that add up to b? • Yes—Rewrite the equation with those factors, group, and factor. • No—You can’t do anything else. If there’s no GCF, it’s a prime polynomial. • No • Go to the next step.

  6. Is it a four-term polynomial? • Yes • Are there two sets of terms that you can group together that have a common factor? • Yes—Group and factor. • No—If it doesn’t have a GCF, it’s a prime polynomial. • No • If it doesn’t have a GCF, it’s a prime polynomial.

  7. NOTE: At EVERY step along the way, you must look at the factors that you get to see if they can be factored any more. Factoring completely means that no factors can be broken down any further using any of the rules you’ve learned.

  8. Practice Factor completely. Is there a GCF? No. Is it a binomial, trinomial, or four-term polynomial? It’s a trinomial. Do you recognize it as a perfect square trinomial? No. Use ac and b.

  9. ac b 1 • 24 14 24 Use your handy-dandy calculator or your super math skills to find 12 and 2 as the factors to use. 12, 2 Rewrite the equation with those two factors in the middle. Group. Factor out the GCF from each group. Write the two factors. Neither one of these factors can be broken down any more, so you’re done.

  10. Factor completely Is there a GCF? Yes. Write the GCF first and the remaining factor after it. Look at the remaining factor. (x-3) Is it a binomial, trinomial, or four-term polynomial? It’s a binomial. Is it a difference of two squares? (a2-b2) No. You can’t do anything else. is the completely factored form.

  11. Factor completely Is there a GCF? Yes. Write the GCF first and the remaining factor after it. Look at the remaining factor. (s2-4) Is it a binomial, trinomial, or four-term polynomial? It’s a binomial. Is it a difference of two squares? (a2-b2) Yes. s2 is a square (s • s) and 4 is a square (2 • 2). Factor as (s+2)(s-2). Then write the complete factorization.

  12. Factor completely Is there a GCF? No. There is no single factor that goes into all four of the terms. Is it a binomial, trinomial, or four-term polynomial? It’s a four-term polynomial. Factor by grouping. Factor out the GCF from each group. Write the two factors.

  13. Factor completely Is there a GCF? Yes. Write the GCF first and the remaining factor after it. Look at the remaining factor. (s2-25t2) Is it a binomial, trinomial, or four-term polynomial? It’s a binomial. Is it a difference of two squares? (a2-b2) Yes. s2 is a square (s • s) and 25t2 is a square (5t • 5t). Factor as (s+5t)(s-5t). Then write the complete factorization.

  14. Factor completely Is there a GCF? No. Is it a binomial, trinomial, or four-term polynomial? It’s a trinomial. Do you recognize it as a perfect square trinomial? No. Use ac and b.

  15. ac b 6 • -6 -5 -36 Look for factors of –36 that add up to –5. Use your calculator or your math skills to find 4 and -9 as the factors to use. 4, -9 Rewrite the equation with those two factors in the middle. Group. Factor out the GCF from each group. Write the two factors.

  16. Factor completely Is there a GCF? Yes. Write the GCF first and the remaining factor after it. Look at the remaining factor. Is it a binomial, trinomial, or four-term polynomial? It’s a trinomial. Do you recognize it as a perfect square trinomial? No. Use ac and b.

  17. ac b 2 • -3 5 -6 Look for factors of -6 that add up to 5. Use your calculator or your math skills to find 6 and -1 as the factors to use. 6, -1 Rewrite the equation with those two factors in the middle. Group. Remember to change the –3 to a +3 because of the minus sign in the grouping!! Factor out the GCF from each group. Write all three factors.

  18. Factor completely Is there a GCF? Yes. Write the GCF first and the remaining factor after it. Look at the remaining factor. Is it a binomial, trinomial, or four-term polynomial? It’s a binomial. Is it a difference of two squares? (a2-b2) Yes. x8 is a square (x4• x4) and 16 is a square (4 • 4). Factor as (x4+ 4)(x4- 4). So far we have 2(x4+ 4)(x4- 4).(Please continue—not done yet!!)

  19. 2(x4+4)(x4-4) Look at what you have. Can either of the binomials be broken down? (x4+4) Is this binomial a difference of two squares? (a2-b2) No. It can’t be broken down. So, we have to keep this factor. What about the other binomial? (x4-4) Is this binomial a difference of two squares? (a2-b2) Yes. x4 is a square (x2• x2) and 4 is a square (2 • 2). Factor as (x2+ 2)(x2- 2).

  20. Put it all together. 2(x4+4)(x4-4) Not a difference of squares. Can’t go any farther!! 2(x4+4)(x2+2)(x2-2)

  21. Word Problem #1 What is the quotient when is divided by 4x? This question is asking you to find the OTHER FACTOR after you take out the greatest common factor of 4x. Simplify each term.

  22. Word Problem #2 A rectangular garden plot has an area represented by the expression Find the dimensions of the garden plot. This is a factoring problem. You need to find the two factors that multiply together to give you

  23. Is there a GCF? No. Is it a binomial, trinomial, or four-term polynomial? It’s a trinomial. Do you recognize it as a perfect square trinomial? No. Use ac and b.

  24. ac b 18 • -28 -3 -504 Look for factors of –504 that add up to –3. Use your calculator or your math skills to find 21 and -24 as the factors to use. 21, -24 Rewrite the equation with those two factors in the middle. Group. Factor out the GCF from each group. Write the two factors. Length is 3x - 4 and width is 6x + 7

  25. Calculator Tips • To find factors of the ac term, use the following steps in your calculator: • Press the Y= button. • In Y1=, type the ac value / X. • In Y2=, type X + VARS, arrow to Y_VARS, Enter, Enter • Go to Table and look for the b in column Y2. When you find it, use the values in the X column and the Y1 column as your two factors to put in the equation. IF YOU CAN’T find the b value in the Y2 column, the trinomial can’t be factored. • NOTE: Remember that you might need to scroll up the screen to find negative numbers that give you the correct value in the Y2 column.

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