Trinomials and inomials Section. Factoring Quadratics Trinomial y 4 13 79 y 4 1 y 8 14 1 inomial y1379 y 4 ( ) 4 3 y The Zero - Product Principle If the product of two algebraic epressions is zero, then at least one of the factors is equal to zero. If A 0, then A 0 or 0 Given ( )( ) 0 Then at least ( ) 0 or ( ) 0 Therefore or and (,0) and (,0) are solutions or zeros 1. If necessary, rewrite the equation in the general form Solving a Quadratic Equation by Factoring a b c 0, moving all terms to one side, thereby obtaining zero on the other side.. Factor completely. 3. Apply the zero-product principle, setting each factor containing a variable equal to zero. 4. Solve the equations in step 3.. Check the solutions in the original equation. The Product of two inomials (sometimes called FOIL) First y ( )( 3) y 3 6 y 6 ( )( 3) Outside ( )(3) 3 Inside ()( ) Last ()(3) 6 (, 0) & ( 3, 0) are solutions/zeros/ -intercepts and (0,6) is the y-intercept In essence you are distributing each term from the first set of parentheses to each term in the second set Factors of 8 (c) Sum of Factors (b) 68 + + 8,1 4,4-8,-1-4,- 9 6-9 -6 Choose either two positive or two negative factors since the sign in front of the 8 is positive. 1
There is no magic bullet to factor trinomials However the following suggestions may help 1 Whenever possible factor out greatest common term first. When the last term is positive, the factors will have the same sign as the middle term. 3 When the last term is negative, the factors will have different signs. 1 Whenever possible factor out greatest common term first 0 116 0 ( 6 8) 0 ( 4)( ) (4,0) and (,0) are solutions or zeros When the last term is positive, the factors will have the same sign as the middle term. 3 When the last term is negative, the factors will have different signs. 0 9 0 0 710 0 616 0 4 1 0 ( )( 4) 0 ( )( ) 0 ( 8)( ) 0 ( 3)( 7) (,0) and (4,0) (,0) and (,0) (8,0) and (,0) (3,0) and ( 7,0) 6 0 3 0 3, or Two numbers that multiply to +6 and add to - 3 0; or 0 Solve each equation (3,0) and (,0) are the two solutions 340 3 40 0 8 0 8 0; or 0 8; or Rewrite Equation in general form Two numbers that multiply to -40 and add to +3 Solve each equation
Possible Factorizations 914 Sum of Inside and Outside products Factoring when a 1 When there is a number in front of the term (ie. a 1) the solution gets more complicated If a= 1, simply factor out the negative value When a 1 or 1 all three coefficients (a, b, and c) must be considered Find two numbers that multiply to the product of a and c and add up to b. Create 4 terms and factor by grouping. 9 - + Possible factorizations Sum of outside and inside products 1 1 1 1 3 3 9 9 3 1 Possible Factorizations Sum of Inside and outside Products Since the sign in front of the is a negative, one factor will be positive and one will be negative. 111 0 Factor Completely 3 0 0; or 3 0 Solve each equation Factoring the Difference of Two Squares ; or 3 ; or 3 3
4 9 3 A A A A, 3 (3)( 3) A A A Memorize this 49 81 7 9 (79)(7 9) A A A A 7, 9 9 36y (3 ) (6 y) (3 6 y)(3 6 y) A A A A 3, 6y 8 3 (a) 8 4 (b) 8 (c) 8 (d) 8 Factoring Perfect Square Trinomials 4
( A A ) ( A ) 136 A ; 6 ( 6)( 6) ( A A ) ( A ) ( 6) ( A A ) ( A ) Memorize these two 16 781 (49)(4 9) (4 9) ( A A ) ( A ) A 4 ; 9 Difference of Two Squares A A A Square of a Sum ( A A ) ( A ) ( A A ) ( A ) A A A A