In this course we will study a number of factoring techniques used to identify the factors of certain polynomials. They are: 1.Greatest Common Factor 2.Grouping 3.Difference of Two Squares 4.Trinomial type 1 (x 2 + bx + c) 5.Trinomial type 2 (ax 2 + bx + c) An important facet in learning how to factor is identifying when to apply each technique. The biggest clue to this is in the number of terms that the original polynomial contains. We will focus on this issue later when we have learned several techniques.

Greatest Common Factor First of all a factor is basically a divisor because all of those numbers can divide evenly into it. Greatest Common Factor for 12 and 18  6 Factors of 12  1, 2, 3, 4, 6, 12 Factors of 18  1, 2, 3, 6,9, 18 Factors of 12  1, 2, 3, 4, 6, 12 Factors of 18  1, 2, 3, 6,9, 18 Common Factors for 12 and 18  1, 2, 3, 6

Just as numbers have factors so do polynomial expressions. m mn + 72n 2 9a 2 – 15ab – 6ac – 6ab b 3 + 4b 2 c 108p 3 q p 3 q + 81p 2 q p 2 q – 5a + 5ab The factors of monomial expressions are pretty obvious, however the factors of polynomial expressions having 2 or more terms are not so obvious. For this reason it is necessary to learn techniques that illuminate the factors of these kinds of polynomial expressions. Monomial expressions Polynomial expressions with 2 or more terms 2x 24a 2 bc 3 108p 2 q

Example 1Factor: 3x 2 + 6xy When determining the GCF of a polynomial, we can take it in steps. 1. Determine the GCF of the numerical coefficients. GCF of 3 and 6 is 3 2. Determine the GCF of each variable. We can do this by choosing the lowest exponent of a given variable from each term. GCF of x 2 and x is x because x is the variable that has the lowest exponent of those 2 (exponent is 1). With no y-variable in the first term and y in the second term there can’t be a common factor containing the variable y.

By dividing each of the terms of the polynomial by this GCF, rewrite the equivalent value of the polynomial as 2 separate factors. One factor being the GCF monomial and the other being the leftover terms after division.

Factor: 3x 2 + 6xy GCF = 3x Factors: 3x(x + 2y) Example 2 Factor: 10x 4 y 3 + 4x 3 y 2 - 2x 2 y 2 GCF of 10, 4 and 2 is 2 GCF of x 4, x 3 and x 2 is x 2 because x 2 is the variable that has lowest exponent of those 3 (exponent is 2). GCF of y 3, y 2 and y 2 is y 2 because y 2 is the variable that has lowest exponent of those 3 (exponent is 2). GCF = 2x 2 y 2

Factor: 10x 4 y 3 + 4x 3 y 2 - 2x 2 y 2 GCF = 2x 2 y 2 Factors: 2x 2 y 2 (5x 2 y + 2x - 1) Example 3Factor: -3m 3 n – 7m 3 r + 8m 3 rt GCF of 3, 7 and 8 is 1 GCF of m 3, m 3 and m 3 is m 3 because each term has the variable m 3. Only one term has the variable n so it is not common to all terms Not all terms have the variable r so it is not common to all terms Only one term has the variable t so it is not common to all terms

GCF = 1m 3 Factors: -m 3 (3n + 7r – 8rt) Factor: -3m 3 n – 7m 3 r + 8m 3 rt Because the leading term (-3m 3 n) is negative, we will make the common factor negative. GCF = -1m 3 Example 4: Factor: -2ab 3 – 4b 3 c – 12b 3 d GCF of 2, 4 and 12 is 2 GCF of b 3, b 3 and b 3 is b 3 because each term has the variable b 3. Only one term has the variable d so it is not common to all terms Not all terms have the variable c so it is not common to all terms Only one term has the variable a so it is not common to all terms

GCF = -2b 3 Factors: -2b 3 (a + 2c + 6d) Because the leading term (-2ab 3 ) is negative, we will make the common factor negative. GCF = -2b 3 Factor: -2ab 3 – 4b 3 c – 12b 3 d

Example 5:Factor:24x 4 y 3 z x 2 y 5 z xz 6 Factors: 4xz 4 (6x 3 y 3 + 4xy 5 – 11z 2 )