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Factoring Trinomials Multiplying Binomials Use Foil Multiply. (x+3)(x+2) Distribute. x2 + 2x + 3x + 6 x2+ 5x + 6 Multiplying Binomials (Tiles) Multiply. (x+3)(x+2) Using Algebra Tiles, we have: x + x x2 3 x x x = x2 + 5x + 6 + x 1 1 1 2 x 1 1 1 Factoring Trinomials (Tiles) How can we factor trinomials such as x2 + 7x + 12 back into binomials? One method is algebra tiles: . 1) Start with x2. x2 x x x x x (vertical or horizontal, at least one of each) and x 1 1 1 1 1 twelve “1” tiles. x 1 1 1 1 1 1 2) Add seven “x” tiles Rearrange until it is a rectangle. 1 Factoring Trinomials (Tiles) How can we factor trinomials such as x2 + 7x + 12 back into binomials? One method is to again use algebra tiles: 1) Start with x2. x2 x x x x x (vertical or horizontal, at least one of each) and x 1 1 1 1 1 twelve “1” tiles. x 1 1 1 1 1 1 2) Add seven “x” tiles 3) Rearrange the tiles until they form a rectangle! 1 We need to change the “x” tiles so the “1” tiles will fill in a rectangle. Factoring Trinomials (Tiles) How can we factor trinomials such as x2 + 7x + 12 back into binomials? One method is to again use algebra tiles: 1) Start with x2. 2) Add seven “x” tiles (vertical or horizontal, at least one of each) and twelve “1” tiles. 3) Rearrange the tiles until they form a rectangle! x2 x x x x x x x 1 1 1 1 1 1 1 1 1 1 1 1 Still not a rectangle. Factoring Trinomials (Tiles) How can we factor trinomials such as x2 + 7x + 12 back into binomials? One method is to again use algebra tiles: 1) Start with x2. x2 x x x (vertical or horizontal, at least one of each) and x 1 1 1 1 twelve “1” tiles. x 1 1 1 1 x 1 1 1 1 2) Add seven “x” tiles 3) Rearrange the tiles until they form a rectangle! A rectangle!!! x Factoring Trinomials (Tiles) How can we factor trinomials such as x2 + 7x + 12 back into binomials? One method is to again use algebra tiles: 4) Top factor: The # of x2 tiles = x’s The # of “x” and “1” columns = constant. 5) Side factor: The # of x2 tiles = x’s The # of “x” and “1” rows = constant. x + 4 x x2 x x x + x 1 1 1 1 3 x 1 1 1 1 x 1 1 1 1 x x2 + 7x + 12 = ( x + 4)( x + 3) Factoring Trinomials Again, we will factor trinomials such as x2 + 7x + 12 back into binomials. look for the pattern of products and sums! If the x2 term has no coefficient (other than 1)... x2 + 7x + 12 Step 1: What multiplies to the last term: 12? 12 = 1 • 12 =2•6 =3•4 Factoring Trinomials x2 + 7x + 12 Step 2: The third term is positive so it must add to the middle term: 7? 12 = 1 • 12 =2•6 =3•4 Step 3: The third term is positive so the signs are both the same as the middle term. Both positive. ( x + 3 )( x + 4 ) x2 + 7x + 12 = ( x + 3)( x + 4) Factoring Trinomials Factor. x2 + 2x - 24 This time, the last term is negative! Step 1: Multiplies to 24. 24 = 1 • 24, Step 2: The third term is negative. That means it subtracts to the middle number and has mixed signs. Step 3: Write the binomial factors and then check your answer. = 2 • 12, = 3 • 8, = 4 • 6, 4 – 6 = -2 6–4=2 x2 + 2x - 24 = ( x - 4)( x + 6) Factoring Trinomials Factor. 3x2 + 14x + 8 This time, the x2 term has a coefficient (other than 1)! Step 1: Multiply 3 • 8 = 24 (the leading coefficient & constant). 24 = 1 • 24 = 2 • 12 Step 2: List all numbers that multiply to 24. Step 3: When the last term is positive the signs are the same. Step 4: Which pair adds up to 14? =3•8 =4•6 Factoring Trinomials 3x2 + 14x + 8 Factor. continued Step 4: Write the factors. Both signs are positive. ( 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 continued You should always check the factors by distributing, especially since this process has more than a couple of steps. ( 3x + 2 )( x + 4 ) = 3x2 + 12x 2x + 8 = 3x2 + 14 x + 8 √ 3x2 + 14x + 8 = (3x + 2)(x + 4) Factoring Trinomials Factor 3x2 + 11x + 4 x2 has a coefficient (other than 1)! Step 1: Multiply 3 • 4 = 12 (the leading coefficient & constant). Step 2: List all the factors of 12. 12 = 1 • 12 =2•6 =3•4 Step 3: Which pair adds up to 11? None If it was 13x, 8x, or 7x, then it could be factored. Because None of the pairs add up to 11, this trinomial can’t be factored; it is PRIME. POP QUIZ! Factor these trinomials: watch your 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: t2 – 4t – 21 1 • 21 3 • 7 3 - 7 or 7 - 3 1) Factors of 21: 2) Which pair subtracts to - 4? 3) Signs are mixed. 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) Solution #3: 1 • 24 2 • 12 3•8 4•6 1) Factors of 32: x2 - 10x + 24 -1 • -24 -2 • -12 -3 • -8 -4 • -6 2) Both signs negative and adds to 10 ? 3) Write the factors. x2 - 10x + 24 = (x - 4)(x - 6) Solution #4: 1) Factors of 18 and subtracts to 3. x2 + 3x - 18 1 • 18 2•9 3•6 2) The last term is negative so the signs are mixed. 3–6=-3 3) Write the factors. -3 + 6 = 3 x2 + 3x - 18 = (x - 3)(x + 6) Solution #5: 1) factors of 42. 2) subtracts to 1 3) Signs are mixed. 4) Put “2” underneath. 2x2 + x - 21 1 • 42 2 • 21 3 • 14 6•7 6 – 7 = -1 7–6=1 ( x - 6)( x + 7) 2 2 3 5) Reduce (if possible). ( x - 6)( x + 7) 2 2 6) Move denominator(s) to the 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) The signs are both positive 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)