A28 APPENDIX Review The first factor to look for in a factoring problem is a common monomial factor present in each term of the polynomial. If one is present, use the Distributive Property to factor it out. Continue factoring out monomial factors until none are left. COMMENT The technique used in part ( ) f is called factoring by grouping. ■ Identifying Common Monomial Factors EXAMPLE 8 Polynomial Common Monomial Factor Remaining Factor Factored Form x2 4 + 2 x 2 + x x 2 4 2 2 ( ) + = + x3 6 − 3 x 2 − x x 3 6 3 2 ( ) − = − x x 2 4 8 2 − + 2 x x2 4 2 − + x x x x 2 4 8 2 2 4 2 2 ( ) − + = − + x8 12 − 4 x2 3 − x x 8 12 4 2 3 ( ) − = − x x 2 + x x 1 + x x x x 1 2 ( ) + = + x x3 3 2 − x2 x 3 − x x x x 3 3 3 2 2( ) − = − x x 6 9 2 + x3 x2 3 + x x x x 6 9 3 2 3 2 ( ) + = + Notice that once all common monomial factors have been removed from a polynomial, the remaining factor is either a prime polynomial of degree 1 or a polynomial of degree 2 or higher. (Do you see why?) The list of special products (2) through (6) given earlier provides a list of factoring formulas when the equations are read from right to left. For example, equation (2) states that if the polynomial is the difference of two squares, x a 2 2 − , it can be factored into x a x a ( )( ) − + . The following example illustrates several factoring techniques. Now Work problems 85, 101, and 135 Factoring Polynomials Factor completely each polynomial. (a) x 16 4 − (b) x 1 3 − (c) x x 9 6 1 2 − + (d) x x4 12 2 + − (e) x x 3 10 8 2 + − (f) x x x 4 2 8 3 2 − + − Solution EXAMPLE 9 (a) x x x x x x 16 4 4 2 2 4 4 2 2 2 ( )( ) ( ) ( )( ) − = − + = − + + ↑ ↑ Difference of squares Difference of squares (b) ( ) ( ) − = − + + x x x x 1 1 1 3 2 ↑ Difference of cubes (c) ( ) − + = − x x x 9 6 1 3 1 2 2 ↑ Perfect square (d) ( )( ) + − = + − x x x x 4 12 6 2 2 ↑ ↑ The sum of 6 and −2 is 4. − = x x x 12 2 10 (e) ( )( ) + − = − + x x x x 3 10 8 3 2 4 2 − x3 8 2 (f) x x x x x x 4 2 8 4 2 8 3 2 3 2 ( ) ( ) − + − = − + − x x x x x 4 2 4 2 4 2 2 ( ) ( ) ( ) ( ) = − + − = + − ↑ ↑ Distributive Property Distributive Property The product of 6 and −2 is −12. ↑ Group terms
RkJQdWJsaXNoZXIy NjM5ODQ=