advertisement

Main Menu Factoring Polynomials A presentation for the greatest Algebra I kids at RJR By Mrs. Sexton Back to last slide Calculator Tips Rules Main Menu Main Menu Rules Step by Step Easy Problems Medium Problems 4my 20m 3 py 15p 8s 2 200t 2 x 2 14x 24 3 2 3x 9 x 2 4s 16 6 y2 5y 6 Hard Problems 6 x3 15x 2 9 x 8 2 x 32 Back to last slide Word Problems Division of polynomial by monomial Find dimensions when area is given Calculator Tips Rules Rules for Factoring Polynomials Back to last slide Calculator Tips Rules Main Menu Step by Step Main Menu • 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. Back to last slide Calculator Tips Rules Main Menu • 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. Back to last slide Calculator Tips Rules Main Menu • 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. Back to last slide Calculator Tips Rules Main Menu 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. Back to last slide Calculator Tips Rules Main Menu Practice Factor completely. x 14x 24 2 No. Is there a GCF? 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. Back to last slide Calculator Tips Rules ac 1 • 24 b 14 24 Main Menu 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. x 14x 24 2 Group. x 12x 2 x 24 2 out the GCF ( x 12x) (2 x 24) Factor from each group. x( x 12) 2( x 12) Write the two factors. ( x 12)(x 2) Neither one of these factors can be 2 broken down any more, so you’re done. Back to last slide Calculator Tips Rules Main Menu Factor completely 3x 9 x 3 2 Is there a GCF? Yes. Write the GCF first and the remaining factor after it. 3x ( x 3) 2 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. 3x ( x 3) 2 Back to last slide is the completely factored form. Calculator Tips Rules Main Menu Factor completely 4s 16 2 Is there a GCF? Yes. Write the GCF first and the remaining factor after it. 4( s 4) 2 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. 4( s 2)(s 2) Back to last slide Calculator Tips Rules Main Menu Factor completely 4my 20m 3 py 15p 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. (4my 20m) (3 py 15 p) Factor out the GCF from each group. 4m( y 5) 3 p( y 5) Write the two factors. (4m 3 p)( y 5) Back to last slide Calculator Tips Rules Main Menu Factor completely 8s 200t 2 2 Is there a GCF? Yes. Write the GCF first and the remaining factor after it. 8(s 25t ) 2 2 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. 8(s 5t )(s 5t ) Back to last slide Calculator Tips Rules Main Menu Factor completely 6y 5y 6 2 No. Is there a GCF? 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. Back to last slide Calculator Tips Rules ac 6 • -6 b -5 Main Menu 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. -36 4, -9 Rewrite the equation with those two factors in the middle. 6y 5y 6 2 6y 9y 4y 6 2 (6 y 9 y) (4 y 6) 2 3 y(2 y 3) 2(2 y 3) Group. Factor out the GCF from each group. Write the two factors. (2 y 3)(3 y 2) Back to last slide Calculator Tips Rules Main Menu Factor completely 6 x 15x 9 x 3 2 Is there a GCF? Yes. Write the GCF first and the remaining factor after it. 3x(2 x 5x 3) 2 Look at the remaining factor. (2x2 5x 3) 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. Back to last slide Calculator Tips Rules ac 2 • -3 b 5 Main Menu 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 6, -1 Rewrite the equation with those two factors in the middle. 3x(2 x 5x 3) 2 3x[2x 6x 1x 3] 2 Group. Remember to change the –3 to a +3 because of the minus sign in the grouping!! 3x[(2x 6x) (1x 3)] 3x[2 x( x 3) 1( x 3)] 3x( x 3)(2 x 1) Factor out the GCF from each group. Write all three factors. 2 Back to last slide Calculator Tips Rules Main Menu Factor completely 2 x 32 8 Is there a GCF? Yes. Write the GCF first and the remaining factor after it. 2( x 16) 8 Look at the remaining factor. ( x8 16) 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!!) Back to last slide Calculator Tips Rules Main Menu 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). Back to last slide Calculator Tips Rules Main Menu Put it all together. 2 x 32 8 2( x 16) 8 2(x4 +4)(x4 -4) 2(x4 +4)(x2 +2)(x2 -2) Back to last slide Calculator Tips Not a difference of squares. Can’t go any farther!! Rules Main Menu Word Problem #1 What is the quotient when 12x3 8x 2 16x is divided by 4x? This question is 3 2 3 2 12 x 8 x 16 x 12 x 8 x 16x asking you to find 4x 4x 4x 4x the OTHER FACTOR after you take out the greatest 2 3x 2x 4 common factor of 4x. Simplify each term. Back to last slide 3x 2 x 4 2 Calculator Tips Rules Main Menu Word Problem #2 A rectangular garden plot has an area represented by the expression 18x 2 3x 28 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 18x 2 3x 28 Back to last slide Calculator Tips Rules Main Menu 18x 3x 28 2 No. Is there a GCF? 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. Back to last slide Calculator Tips Rules ac 18 • -28 b -3 Main Menu 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. -504 21, -24 Rewrite the equation with those two factors in the middle. 18x 3x 28 2 18x 24x 21x 28 2 (18x 24x) (21x 28) 2 6 x(3x 4) 7(3x 4) (3x 4)(6 x 7) Back to last slide Group. Factor out the GCF from each group. Write the two factors. Length is 3x - 4 and width is 6x + 7 Calculator Tips Rules Main Menu 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. Back to last slide Calculator Tips Rules