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Factoring Polynomials A polynomial is a sum or subtraction of monomials i e 10x2 15x is a polynomial and so is x3 3x2 5x 9 In any factorization problem the first thing to look at is the greatest common factor Factoring out the Greatest Common Factor GCF 18x3 27x2 In this polynomial 9 is the greatest integer that divides 18 and 27 x2 is the greatest expression that divides x3 and x2 9x2 2x 9x2 3 9x2 2x 3 Some polynomials may have a GCF of 1 but appropriate grouping may lead to possible factorization Factoring by Grouping x3 4x2 3x 12 x3 4x2 3x 12 Common factor in x3 4x2 is x2 and common factor in 3x 12 is 3 These can be factored as x2 x 4 3 x 4 x 4 x2 3 Factoring Trinomials A strategy for factoring ax2 bx c 1 Find 2 numbers whose product is ac and whose sum gives b Say the numbers are u and v 2 Re write the trinomial such as ax2 ux vx c 3 Use factoring by grouping to find the factors Here is an example x2 3x 18 We need to find two numbers such that their product is 18 and their sum is 3 Let s find the factors of 18 Factors of 18 Sum of factors 18 1 17 Created by UASP Student Success Centers success asu edu 480 965 9072 18 1 17 9 2 7 9 2 7 6 3 3 6 3 3 Factoring Polynomials Our desired numbers are 6 and 3 We will now re write our trinomial x2 6x 18 x2 6x 3x 18 x x 6 3 x 6 x 3 x 6 Let s look at another example whose leading coefficient is not 1 8x2 10x 3 We need to find two numbers such that their product is 24 and their sum is 10 Let s find the factors of 24 Factors of 24 Sum of Factors 24 1 23 24 1 23 12 2 10 12 2 10 Our desire numbers are 12 and 2 We will now re write our trinomial 8x2 12x 2x 3 8x2 12x 2x 3 4x 2x 3 1 2x 3 4x 1 2x 3 Created by UASP Student Success Centers success asu edu 480 965 9072 8 3 5 8 3 5 6 4 2 6 4 2 Factoring Polynomials A Strategy for Factoring a Polynomial 1 If there is a common factor factor out the GCF 2 Determine the number of terms in the polynomial and try factoring as follows a If there are two terms then can the binomial be factored