Machinery's Handbook, 31st Edition
30 FACTORING POLYNOMIALS Example : Three similar trinomials, only two are factorable: x x x x x x x 2 2 2 3 3 1 3 1 3 3 1 2 2 3 + = + − = + = = − − − − − − − ( )( ), )( ) ( since ( and 3 1 3 1 3 3 1 2 2 3 2 )( ), ( )( ) x x x + = + = + − − − − − since and Examples : Factoring by reverse FOIL, along with the explanation: x 2 – 8 x – 20 = ( x – 10)( x + 2) The two integers whose product is –20 and whose sum is –8 are –10 and 2. x 2 – 21 x + 20 = ( x – 20)( x – 1) The two integers whose product is 20 and whose sum is –21 are –20 and –1. Does not factor over the real numbers, since no two integers have a product of 3 and a sum + of − 2 Reverse FOIL for polynomials of Form ax 2 + bx + c : If the leading coefficient is other than 1, reverse FOIL works, but a different tactic is needed. Sometimes called “magic factoring,” the procedure is shown in this example: 3 x 2 + 13 x + 12, a = 3, b = 13, c = 12 Step 1: ac = (3)(12) = 36 Step 2 : Write all factor pairs of ac = 36: 1, 36; 2, 18; 3, 12; 4, 9; 6, 6 Step 3 : Choose the factor pair whose sum is b = 13: 4, 9 Step 4 : Rewrite polynomial with middle term 13 x as 4 x + 9 x : 3 x 2 + 4 x + 9 x + 12 Step 5 : Group first two terms and second two terms in parentheses: (3 x 2 + 4 x ) + (9 x + 12) Step 6 : Take the common factor out of each group: x (3 x + 4) + 3(3 x + 4) Step 7 : Take the common binomial factor out of each large term: (3 x + 4)( x + 3) Correct factorization Step 8 : Check the factorization found: 3 x 2 + 9 x + 4 x + 12 = 3 x 2 + 13 x + 12 Special Products : Certain binomials are factored according to formulas, which can be checked by multiplying. The difference of squares is perhaps the most important because it comes up so often in applications, as do many quadratic (second-degree) polynomials. It comes from multiplying the conjugate pair of binomials, ( a + b ) with ( a – b ) to get a 2 – b 2 . Difference of squares: a 2 – b 2 = ( a + b )( a – b ) Difference of cubes: a 3 – b 3 = ( a – b )( a 2 + ab + b 2 ) Sum of cubes: a 3 + b 3 = ( a + b )( a 2 – ab + b 2 ) Square of a sum: ( a + b ) 2 = ( a + b )( a + b ) = a 2 + 2 ab + b 2 Difference of a sum: ( a – b ) 2 = ( a – b )( a – b ) = a 2 – 2 ab + b 2 Note : The sum of squares a 2 + b 2 is not a factorable binomial over the set of real num- bers . However, it is factorable over the set of complex numbers (see Complex Numbers on page 59). Examples : Several special product factorizations are shown: Difference of squares: Differerence of cubes: S a b a b 2 2 3 3 − − um of cubes: a b x x x x x x x x 3 3 2 3 2 3 81 9 9 1 1 1 + − = + − − = − + + + ( )( ) ( )( ) 1 1 1 10049 107107 8 2 4 2 2 2 3 2 = + + − = + − − = − + − + ( )( ) ( )( ) ( )( ) x x x x x x x x x x 27 3 9 3 3 2 + = + + x x x x ( )( ) – Difference of squares: Differerence of cubes: S a b a b 2 2 3 3 − − um of cubes: a b x x x x x x x x 3 3 2 3 2 3 81 9 9 1 1 1 + − = + − − = − + + ( )( ) ( )( ) 1 1 1 10049 107107 8 2 4 2 2 2 3 2 = + + − = + − − = − + − + ( )( ) ( )( ) ( )( ) x x x x x x x x x x 27 3 9 3 3 2 + = + + x x x x ( )( ) – Difference of squares: Differerence of cubes: S a b a b 2 3 3 − − um of cubes: a b x x x x x x x x 3 3 2 3 2 3 81 9 9 1 1 1 + − = + − − = − + + + ( )( ) ( )( ) 1 1 1 10049 107107 8 2 4 2 2 2 3 2 = + + − = + − − = − − + ( )( ) ( )( ) ( )( ) x x x x x x x x x 27 3 9 3 3 2 + = + + x x x ( )( ) – Factorization of all other cases is done using the quadratic formula , as shown in the next section.
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