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FACTORING Objective: To factor polynomials using a variety of methods Definitions: Product: The result of a multiplication problem Factor: The numbers and/or terms that are being multiplied Factoring: The process of breaking a number or polynomial into its smallest factors Types of Factoring Greatest Common Factor Difference of Squares Perfect Square Trinomial Factoring a trinomial into two binomials (using guess and check) Sum/Difference of Cubes Greatest Common Factor The GCF is the largest monomial that will divide evenly into EVERY term of the polynomial You should always look for a GCF first when doing any factoring problem Ex. In 20x4 + 35x2 the GCF is 5x2 In 32a3b5 – 24a2b7 – 40ab8 the GCF is 8ab5 When checking for a GCF, put the polynomial in standard form. If the leading coefficient is negative, then the GCF should be negative. Greatest Common Factor Once you have determined the GCF, you will divide it out of each term. The result should be written as GCF(polynomial result with GCF factored out) Example: 20x4 + 35x2 GCF = 5x2 20x 4 35x2 2 4x 7 2 5x 20x4 + 35x2 = 5x2(4x2 +7) Greatest Common Factor Example: 6a3b5 – 9a4b6 + 15ab8 – 18a2b7 Put in standard form: GCF: -9a4b6 + 6a3b5 – 18a2b7 + 15ab8 -3ab5 Factored form: -3ab5(3a3b – 2a2 + 6ab2 – 5b3) Example: 12m4 – 16m3 + 24m2 GCF: 4m2 Factored form: 4m2(3m2 – 4m + 6) Factoring a Difference of Perfect Squares Multiply: (2x – 3)(2x + 3) 4x2 + 6x – 6x – 9 4x2 – 9 4x2 – 9 is known as a Difference of Perfect Squares A Difference of Perfect Squares is a BINOMIAL where both terms are perfect squares and the terms are subtracted. A Difference of Perfect Squares a2 – b2 is factored into two binomials: (a + b)(a – b) Factoring a Difference of Perfect Squares Example: 16m2 – 25 (4m + 5)(4m – 5) Example: 18a3 – 98a GCF: 2a Factor: 2a(9a2 – 49) This is a difference of squares Final factored form: 2a(3a + 7)(3a – 7) Factoring a Perfect Square Trinomial Multiply: (2x + 3)2 (2x + 3)(2x + 3) 4x2 + 6x + 6x + 9 4x2 + 12x + 9 A Perfect Square Trinomial is a TRINOMIAL where the first and last terms are perfect squares and the middle term is twice the product of the square roots of the first and last terms (in the example above the square root of the first term is 2x, the square root of the last term is 3, and the middle term is equal to 2 2x 3 or 12x). The general form of a Perfect Square Trinomial is either a2 + 2ab + b2 or a2 – 2ab + b2 and factors to either (a + b)2 or (a – b)2 Factoring a Perfect Square Trinomial Factor: x2 + 10x + 25 The first and last terms are perfect squares, and the middle term is double the product of the square roots of the first and last terms. Factored form: Factor: (x + 5)2 9x2 – 24x + 16 The first and last terms are perfect squares (3x and 4) and the middle term is double their product (2 12x) Factored form: Factor: (3x – 4)2 25x2 – 30x – 9 The first and last terms are perfect squares, but the last term is negative so this IS NOT a Perfect Square Trinomial and cannot be factored using the rules for a Perfect Square Trinomial Factoring a General Trinomial Multiply: (x + 3)(x – 5) x2 – 5x + 3x – 15 x2 – 2x – 15 (x + 3) and (x – 5) are the factors of x2 - 2x – 15 When factoring a trinomial, first check to see if there is a GCF or if it is a Perfect Square Trinomial. If not, the easiest method is to guess and check to find the factors. Factoring a General Trinomial Example: Factor x2 – 4x + 3 Because the leading coefficient is 1, we can look for two numbers whose product is 3 and whose sum is -4. Those two numbers are -3 and -1 The two binomials whose product is x2 – 4x + 3 (factors) are (x – 3) and (x – 1) The factored form is (x – 3)(x – 1) Example: Factor x2 + 6x – 16 Because the leading coefficient is 1, we can look for two numbers whose product is -16 and whose sum is 6. Those two numbers are 8 and -2. The factored form is (x + 8)(x – 2) Factoring a General Trinomial Example: Factor 2x2 + 7x – 15 Because the leading coefficient is not 1, this is a little more difficult to factor. The first terms of each binomial factor must multiply to be 2x2, and the second terms of each binomial factor must multiply to be -15. Guess and check different combinations until you find one that works. Guess #1: (2x – 5)(x + 3) This is not correct! Check: 2x2 + 6x – 5x – 15 = 2x2 + x – 15 Guess #2: (2x + 3)(x – 5) This is not correct! Check: 2x2 – 10x + 3x – 15 = 2x2 – 7x – 15 Guess #3: (2x – 3)(x + 5) YEA!!! We have found the correct factors! Check: 2x2 + 10x – 3x – 15 = 2x2 + 7x - 15 Factoring a General Trinomial Example: Factor 6x2 + 11x - 10 The first terms in each binomial factor must multiply to be 6x2 and the last terms in each binomial factor must multiply to be -10. Guess and check until you find the combination that yields the correct middle term. (2x + 5)(3x – 2) Check: Example: 6x2 – 4x + 15x – 10 = 6x2 + 11x – 10 Factor 10x3 + 35x2 + 15x There is a GCF of 5x 5x(2x2 + 7x + 3) Guess and check until you find the right binomials to factor the trinomial 2x2 + 7x + 3 factors to be (2x + 1)(x + 3) Final factored form: 5x(2x + 1)(x + 3) Factoring a Sum or Difference of Cubes A Sum or Difference of Cubes is a BINOMIAL where each term is a perfect cube. The Sum of Cubes a3 + b3 factors into a binomial multiplied by a trinomial with the pattern: (a + b)(a2 – ab + b2) The Difference of Cubes a3 – b3 factors into a binomial multiplied by a trinomial with the pattern: (a – b)(a2 + ab + b2) Factoring a Sum or Difference of Cubes Example: Factor 27x3 + 125 This is a sum of cubes (3x and 5) The factors are (3x + 5)(9x2 – 15x + 25) Example: Factor 16x4y – 54xy4 There is a GCF of 2xy Factoring out the GCF yields 2xy(8x3 – 27y3) This is a difference of cubes! The final factored form is 2xy(2x – 3y)(4x2 + 6xy + 9y2)