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Factoring Trinomials

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Multiply. (x+3)(x+2) x 2 + 2x + 3x + 6 Multiplying Binomials Use Foil x 2 + 5x + 6 Distribute.

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x + 3 x+2x+2 Using Algebra Tiles, we have: = x 2 + 5x + 6 Multiplying Binomials (Tiles) Multiply. (x+3)(x+2) x2x2 x x 1 xx x 11111

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How can we factor trinomials such as x 2 + 7x + 12 back into binomials? One method is algebra tiles:. 1) Start with x 2. Factoring Trinomials (Tiles) 2) Add seven x tiles (vertical or horizontal, at least one of each) and twelve 1 tiles. x2x2 xxxxx x x Rearrange until it is a rectangle.

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How can we factor trinomials such as x 2 + 7x + 12 back into binomials? One method is to again use algebra tiles: 1) Start with x 2. Factoring Trinomials (Tiles) 2) Add seven x tiles (vertical or horizontal, at least one of each) and twelve 1 tiles. x2x2 xxxxx x x ) Rearrange the tiles until they form a rectangle! We need to change the x tiles so the 1 tiles will fill in a rectangle.

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How can we factor trinomials such as x 2 + 7x + 12 back into binomials? One method is to again use algebra tiles: 1) Start with x 2. Factoring Trinomials (Tiles) 2) Add seven x tiles (vertical or horizontal, at least one of each) and twelve 1 tiles. x2x2 xxxxx x ) Rearrange the tiles until they form a rectangle! Still not a rectangle. x

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How can we factor trinomials such as x 2 + 7x + 12 back into binomials? One method is to again use algebra tiles: 1) Start with x 2. Factoring Trinomials (Tiles) 2) Add seven x tiles (vertical or horizontal, at least one of each) and twelve 1 tiles. x2x2 xxxx x ) Rearrange the tiles until they form a rectangle! A rectangle!!! x x

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How can we factor trinomials such as x 2 + 7x + 12 back into binomials? One method is to again use algebra tiles: 4) Top factor: The # of x 2 tiles = xs The # of x and 1 columns = constant. Factoring Trinomials (Tiles) 5) Side factor: The # of x 2 tiles = xs The # of x and 1 rows = constant. x2x2 xxxx x x 2 + 7x + 12 = ( x + 4)( x + 3) x x x + 4 x +3+3

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Again, we will factor trinomials such as x 2 + 7x + 12 back into binomials. look for the pattern of products and sums! Factoring Trinomials If the x 2 term has no coefficient (other than 1)... Step 1: What multiplies to the last term: 12? x 2 + 7x = 1 12 = 2 6 = 3 4

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Factoring Trinomials Step 2: The third term is positive so it must add to the middle term: 7? x 2 + 7x = 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 + )( x + ) 3 4 x 2 + 7x + 12 = ( x + 3)( x + 4)

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Factor. x 2 + 2x - 24 This time, the last term is negative! Factoring Trinomials Step 1: Multiplies to = 1 24, = 2 12, = 3 8, = 4 6, 4 – 6 = -2 6 – 4 = 2 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. x 2 + 2x - 24 = ( x - 4)( x + 6)

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Factor. 3x x + 8 This time, the x 2 term has a coefficient (other than 1)! Factoring Trinomials Step 2: List all numbers that multiply to = 1 24 = 2 12 = 3 8 = 4 6 Step 4: Which pair adds up to 14? Step 1: Multiply 3 8 = 24 (the leading coefficient & constant). Step 3: When the last term is positive the signs are the same.

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( 3x + 2 )( x + 4 ) 2 Factor. 3x x + 8 continued Factoring Trinomials Step 5: Put the original leading coefficient (3) under both numbers. ( x + )( x + ) Step 6: Reduce the fractions, if possible. Step 7: Move denominators in front of x. Step 4: Write the factors. Both signs are positive ( x + )( x + ) ( x + )( x + ) 4 3

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

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Factor 3x x + 4 x 2 has a coefficient (other than 1)! Factoring Trinomials Step 2: List all the factors of = 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. Step 1: Multiply 3 4 = 12 (the leading coefficient & constant). Because None of the pairs add up to 11, this trinomial cant be factored; it is PRIME.

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Factor these trinomials: watch your signs. 1) t 2 – 4t – 21 2) x x ) x 2 –10x ) x 2 + 3x – 18 5) 2x 2 + x – 21 6) 3x x + 10 POP QUIZ!

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Solution #1: t 2 – 4t – 21 1) Factors of 21: or ) Which pair subtracts to - 4? 3) Signs are mixed. t 2 – 4t – 21 = (t + 3)(t - 7)

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

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Solution #3: x x ) Factors of 32: ) Both signs negative and adds to 10 ? 3) Write the factors. x x + 24 = (x - 4)(x - 6)

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Solution #4: x 2 + 3x ) Factors of 18 and subtracts to ) The last term is negative so the signs are mixed. 3) Write the factors. x 2 + 3x - 18 = (x - 3)(x + 6) 3 – 6 = = 3

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Solution #5: 2x 2 + x ) factors of – 7 = -1 7 – 6 = 1 2) subtracts to 1 3) Signs are mixed. 2x 2 + x - 21 = (x - 3)(2x + 7) ( x - 6)( x + 7) 4) Put 2 underneath ) Reduce (if possible). ( x - 6)( x + 7) ) Move denominator(s) to the front of x. ( x - 3)( 2x + 7)

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Solution #6: 3x x ) Multiply 3 10 = 30; list factors of ) Which pair adds to 11 ? 3) The signs are both positive 3x x + 10 = (3x + 5)(x + 2) ( x + 5)( x + 6) 4) Put 3 underneath ) Reduce (if possible). ( x + 5)( x + 6) ) Move denominator(s)in front of x. ( 3x + 5)( x + 2)

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