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10-5 Factoring Quadratic Trinomials Homework Any Questions? Factoring quadratic trinomials is FUN!! It is kind of like solving a mathematical riddle or puzzle. When you factor a trinomial there are 2 clues to take into account. 1. Operator Signs Trinomial Binomial + + ( + )( + ) - + ( -)( - ) - - ( - )( + ) + - ( + )( - ) •The first sign remains the SAME the second sign is the result of multiplying the signs together (a positive # x a positive # = a positive #; a negative # x a positive # = a negative #...) 2. Factors of c that add/subtract to equal b Example 1: Factor the trinomial x2 + 3x + 2 Step 1: Check the signs Step 2: Factors of “c” that add up to “b” Step 3: Factor your trinomial to 2 binomials Step 4: Check with FOIL Let’s try the steps… Step 1: + + (+)(+) Step 2: Factors of 2 are 2 & 1 and 2 + 1 = 3 Step 3: (x + 2)(x + 1) Step 4: x2 + 2x + x + 2 , combine like terms to get… x2 + 3x + 2 Cool Fact of the Day Some species of fish have voices. 5 Example 2: Factor the trinomial x2 – 2x – 8 Step 1: (-)(+) Step 2: Factors of -8 which add up to 2 are -4 & 2 Step 3: (x – 4) (x + 2) Step 4: x2 + 2x – 4x – 8 x2 – 2x – 8 original trinomial Student of the Day Factoring ax2 + bx + c when a = 1 What two numbers multiply to give you 8, but add to give you 6? To FACTOR a trinomial means to write it as the product of two binomials. Factor x2 + 6x + 8 What twotwo numbers Write sets of multiply to give you parenthesis and the last number… filltoin the and add give you the middle number? numbers. (x + 2)) (x + 4)) Ex: 1 Factor x2 - 3x + 2 (x - 2)) (x - 1)) What two numbers multiply to give you the last number… and add to give you the middle number? Ex: 2 Factor x2 - 2x - 8 (x - 4) (x + 2) Ex: 3 Factor x2 - 5x - 14 (x - 7)) (x + 2)) Ex: 4 Factor x2 - 16x + 64 (x - 8)) (x - 8)) Same thing 2 as (x - 8) Ex: 5 Factor x2 - x - 42 (x + 6)) (x - 7)) (x Solve the equation by factoring Ex: 6 x2 - 8x + 12 = 0 (x )(x )=0 x-6=0 or x–2=0 x=6 or x =2 Solve the equation by factoring 2 Ex: 7 x - 7 = - 6x x2 + 6x - 7 = 0 (x )(x )=0 x-1=0 or x+7=0 x=1 or x = -7 Homework Complete Sum and Product Puzzles plus Pg. 539 #5-19 odd Now, factoring can become a little more challenging when “a” is not 1. You will use the strategy of Guess and Check to find the factors. Your goal is to find a combination of factors of a and c so that the inner and outer products add to the middle term of bx. Factors of “a” ax2 + bx + c = ( _x + _ )( _x + _ ) Factors of “c” Example 3: Factor the Trinomial: 6x2 – 7x – 5 You will have to use Guess and Check for this trinomial because “a” is not one anymore. Guess & Check (x + 1)(6x – 5) 6x2 + x – 5, does not work (x + 5)(6x – 1) 6x2 + 29x – 5 (2x + 1)(3x – 5) 6x2 – 7x – 5, yes we have found the answer! Example 4: Factor the Trinomial: 6x2 – 7x – 5 New method 1. (6x )(6x 2. (6x – )(6x 3. (6x– )(6x + ) First term, without the exponent, goes in each binomial ) First sign goes in the first binomial ) Both signs multiplied together, goes in the second binomial (negative times a negative equals a positive) 4. Multiply a and c together = 30 then find all the factor combinations for 30… 1,30 2,15 3,10 5,6 5. If the signs in the two binomials are the same use the two factors that add to b If the signs in the two binomials are different use the two factors that subtract to b 6. (6x–10)(6x +3) Put the larger factor in the first binomial and the other in the second binomial. 7. Factor each binomial (6x–10)/2 (6x +3)/3 (3x-5) (2x+1) Student of the Day Many trinomials cannot be factored with integer coefficients and there is a way to check for factorability. Using the discriminant, b2 – 4ac, to check for factorability. If your answer is a perfect square, it can be factored, but if it is not a perfect square then you must use other methods. Example 4: 2x2 + 3x – 6 b2 – 4ac = 32 – 4 (2)(-6) = 57 Therefore, this trinomial cannot be factored. Cool Photo of the Day New Invention 23 Cool Photo of the Day New Invention 24 Cool Photo of the Day New Invention 25 Cool Photo of the Day New Invention 26 Factor the trinomial. Use the discriminant to decide whether the polynomial can be factored with integer coefficients. If it can, factor it. Homework Homework: Pg. 539 29-33 odd Pg. 556 #37-46