Factoring Trinomials

Factoring Trinomials

Factoring Trinomials We will factor trinomials such as x2 7x 12 back into binomials. If the x2 term has no coefficient (other than 1). x2 7x 12 Step 1: List all pairs of numbers that multiply to equal the constant, 12. 12 1 12 2 6 3 4

Factoring Trinomials x2 7x 12 Step 2: Choose the pair that adds up to the middle coefficient. 12 1 12 2 6 3 4 Step 3: Fill those numbers into the blanks in the binomials: ( x 3 )( x 4 ) x2 7x 12 ( x 3)( x 4)

Factoring Trinomials Factor. x2 2x - 24 This time, the constant is negative! Step 1: List all pairs of numbers that multiply to equal the constant, -24. (To get -24, one number must be positive and one negative.) -24 1 -24, -1 24 2 -12, -2 12 3 -8, -3 8 4 -6, - 4 6 Step 2: Which pair adds up to 2? Step 3: Write the binomial factors. x2 2x - 24 ( x - 4)( x 6)

Factor These Trinomials! Factor each trinomial, if possible. The first four do NOT have leading coefficients, the last two DO have leading coefficients. Watch out for signs!! 1) t2 – 4t – 21 2) x2 12x 32 3) x2 –10x 24 4) x2 3x – 18 5) 2x2 x – 21 6) 3x2 11x 10

Solution #1: 1) Factors of -21: t2 – 4t – 21 1 -21, -1 21 3 -7, -3 7 2) Which pair adds to (- 4)? 3) Write the factors. t2 – 4t – 21 (t 3)(t - 7)

Solution #2: x2 12x 32 1 32 2 16 4 8 1) Factors of 32: 2) Which pair adds to 12 ? 3) Write the factors. x2 12x 32 (x 4)(x 8)

x2 - 10x 24 Solution #3: 1 24 2 12 3 8 4 6 1) Factors of 32: 2) Which pair adds to -10 ? -1 -24 -2 -12 -3 -8 -4 -6 None of them adds to (-10). For the numbers to multiply to 24 and add to -10, they must both be negative! 3) Write the factors. x2 - 10x 24 (x - 4)(x - 6)

Solution #4: 1) Factors of -18: x2 3x - 18 1 -18, -1 18 2 -9, -2 9 3 -6, -3 6 2) Which pair adds to 3 ? 3) Write the factors. x2 3x - 18 (x - 3)(x 18)

Factoring Trinomials Factor. 3x2 14x 8 This time, the x2 term DOES have a coefficient (other than 1)! Step 1: Multiply 3 8 24 (the leading coefficient & constant). 24 1 24 2 12 Step 2: List all pairs of numbers that multiply to equal that product, 24. Step 3: Which pair adds up to 14? 3 8 4 6

Factoring Trinomials Factor. 3x2 14x 8 Step 4: Write temporary factors with the two numbers. ( x 2 )( x 12 ) 3 3 Step 5: Put the original leading coefficient (3) under both numbers. ( x 2 )( x 12 ) 3 3 Step 6: Reduce the fractions, if possible. ( x 2 )( x 4 ) 3 Step 7: Move denominators in front of x. ( 3x 2 )( x 4 ) 4

Factoring Trinomials Factor. 3x2 14x 8 You should always check the factors by distributing, especially since this process has more than a couple of steps. ( 3x 2 )( x 4 ) 3x x 3x 4 2 x 2 4 3x2 14 x 8 3x2 14x 8 (3x 2)(x 4)

Factoring Trinomials Factor 3x2 11x 4 This time, the x2 term DOES have a coefficient (other than 1)! Step 1: Multiply 3 4 12 (the leading coefficient & constant). Step 2: List all pairs of numbers that multiply to equal that product, 12. 12 1 12 2 6 3 4 Step 3: Which pair adds up to 11? None of the pairs add up to 11, this trinomial can’t be factored; it is PRIME.

Factor These Trinomials! Factor each trinomial, if possible. The first four do NOT have leading coefficients, the last two DO have leading coefficients. Watch out for signs!! 1) t2 – 4t – 21 2) x2 12x 32 3) x2 –10x 24 4) x2 3x – 18 5) 2x2 x – 21 6) 3x2 11x 10

Solution #5: 1) Multiply 2 (-21) - 42; list factors of - 42. 2) Which pair adds to 1 ? 3) Write the temporary factors. 4) Put “2” underneath. 2x2 x - 21 1 -42, -1 42 2 -21, -2 21 3 -14, -3 14 6 -7, -6 7 ( x - 6)( x 7) 2 2 3 5) Reduce (if possible). ( x - 6)( x 7) 2 2 6) Move denominator(s)in front of “x”. ( x - 3)( 2x 7) 2x2 x - 21 (x - 3)(2x 7)

Solution #6: 1) Multiply 3 10 30; list factors of 30. 2) Which pair adds to 11 ? 3) Write the temporary factors. 4) Put “3” underneath. 3x2 11x 10 1 30 2 15 3 10 5 6 ( x 5)( x 6) 3 3 2 5) Reduce (if possible). ( x 5)( x 6) 3 3 6) Move denominator(s)in front of “x”. ( 3x 5)( x 2) 3x2 11x 10 (3x 5)(x 2)