Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16R.5 Factoring PolynomialsChapter R Section 5: Factoring PolynomialsIn 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 Cubes2x bx c+ +2ax bx c+ +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 polynomialR.5 Factoring Polynomials: Factor Out the GCF( )3 2 1x x -26 3x x-We are, in essence, un-distributingExamples: Factor out the GCF (greatest common factor).R.5 Factoring Polynomials: Factor Out the GCF2 2 360 48 72x y xy x y- +5(3 7) (3 7)x x x- + -Examples: Factor by grouping.R.5 Factoring Polynomials: Factor by Grouping26 9 4 6x x x+ + +3 24 2 8x x x- + -multiply the polynomials factor the polynomialFactoring 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.R.5 Factoring Polynomials: Factor Trinomials of the Form x + bx + c22 5 6x x+ +( 2)( 3)x x+ +We are, in essence, un-FOIL-ingsum productExamples: Factor each trinomial.216 10x x+ -212 36y y- +R.5 Factoring Polynomials: Factor Trinomials of the Form x + bx + c2Some Handy-Dandy Special Product FormulasPerfect SquaresDifference of Two Squaressome perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144,…( )22 22x ax a x a+ + = +( )22 22x ax a x a- + = -( ) ( )2 2x a x a x a- = + -Examples: Factor each trinomial29x -29 64y -R.5 Factoring Polynomials: Factor Trinomials of the Form x + bx + c2R.5 Factoring Polynomials: Factor Trinomials of the Form ax + bx + c2Factoring Trinomials with a leading coefficient other than 1 (2 methods) 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!) 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 grouping26 7 3x x+ -26 7 3x x+ -R.5 Factoring Polynomials: Factor Trinomials of the Form ax + bx + c225 4 1x x+ +2 218 3 10x xy y- -Examples: Factor each trinomial.28 10 3x x- +24 9 9a a- -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. 2ax bx c+ + 0a �24b ac-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 askedAlgebra:R.5 Factoring Polynomials: Factor Trinomials of the Form ax + bx + c2 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. 28 10 3x x- +24b ac-Y= GRAPH 2ND TRACECheck to see if the factors are correct!R.5 Factoring Polynomials: Factor the Sum and Difference of Two Perfect CubesSpecial Product Formulas Sum of Two Cubes Difference of Two Cubes some perfect cubes: 1, 8, 27, 64, 125, …( )( )3 3 2 2x a x a x ax a- = - + +( )( )3 3 2 2x a x a x ax a+ = + - +38x +327 64x -Examples: Factor the sum or difference of two cubes.‘sopR.5 Factoring PolynomialsFactor-ific Flow Chart – Cut off this slide and use as a reference.Is there a common factor?Is the leading coefficient negative?Write the polynomial in descending order (for the first variable alphabetically)How many terms are there?Determine if it a difference of squares, difference of cubes or sum of cubes?Try factoring by trial-and-error (or use key number method)Try factoring by grouping.Check each of the factors to see if we can factor further. Factor out a -1 (or –GCF)Factor out the GCFnonoyesyesthreetwofourQuestion 2Question 3Question 2Question 3Examples: Factor each polynomial completely.R.5 Factoring Polynomials41x -29 9 4b b+ -24 32x x- + +23 27x-Examples: Factor each polynomial completely.R.5 Factoring Polynomials24 4x y xy x+ - -364 27x+8 5x x-24 8 32a a- +R.5 Factoring PolynomialsIndependent 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. 49-55Homework: pp. 56-57 #9-21 odds, 25, 27, 29, 33, 35, 37, 43-53 odds, 59-71 odds, 79-107 oddsPretend 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