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Factoring Trinomials

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Multiply (x+3)(x+2) Multiplying Binomials (FOIL) F O I L = x 2 + 2x + 3x + 6 = 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 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

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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. This method does not use tiles, instead we look for the pattern of products and sums! Factoring Trinomials ( Method 2 ) If the x 2 term has no coefficient (other than 1)... Step 1: List all pairs of numbers that multiply to equal the constant, 12. x 2 + 7x = 1 12 = 2 6 = 3 4

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

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Factor. x 2 + 2x - 24 This time, the constant is negative! Factoring Trinomials ( Method 2 ) 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, = 2 -12, = 3 -8, -3 8 = 4 -6, Step 2: Which pair adds up to 2? Step 3: Write the binomial factors. x 2 + 2x - 24 = ( x - 4)( x + 6)

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Factor. 3x x + 8 This time, the x2 x2 term DOES have a coefficient (other than 1)! Factoring Trinomials ( Method 3* ) Step 2: List all pairs of numbers that multiply to equal that product, = 1 24 = 2 12 = 3 8 = 4 6 Step 3: Which pair adds up to 14? Step 1: Multiply 3 8 = 24 (the leading coefficient & constant). a) Factor by Decomposition Step 4: Rewrite the x term as two terms Step 5: Divide the new polynomial into two pairs and factor out the common factor from each pair.

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Factor. 2x 2 - 7x -15 This time, the x 2 term DOES have a coefficient (other than 1)! Factoring Trinomials ( Method 2* ) Step 2: List all pairs of numbers that multiply to equal that product, Step 3: Which pair adds up to ? Step 1: Multiply (the leading coefficient & constant). a) Factor by Decomposition Step 4: Rewrite the x term as two terms Step 5: Divide the new polynomial into two pairs and factor out the common factor from each pair.

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

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( 3x + 2 )( x + 4 ) 2 Factor. 3x x + 8 Factoring Trinomials ( Method 2* ) 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 temporary factors with the two numbers ( x + )( x + ) ( x + )( x + ) 4 3

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

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Factor 3x x + 4 This time, the x 2 term DOES have a coefficient (other than 1)! Factoring Trinomials ( Method 2* ) Step 2: List all pairs of numbers that multiply to equal that product, = 1 12 = 2 6 = 3 4 Step 3: Which pair adds up to 11? Step 1: Multiply 3 4 = 12 (the leading coefficient & constant). None of the pairs add up to 11, this trinomial cant be factored; it is PRIME.

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Factor 4x This time, the Quadratic Equation is a Binomial with a Subtraction Sign! Factoring A Difference of Squares (Method 3) Step 2: List two brackets, one with an addition sign, one with a subtraction sign. Step 3: Place the square roots in each bracket. Step 1: Check to see that both terms are perfect squares. Check your answer by expanding! 2x 4 4

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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) x 2 – 4x– 21 2) x x ) x 2 –10x ) x 2 + 3x – 18 5) 2x 2 + x – 21 6) 3x x + 10 Factor These Trinomials!

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Solution #1: x 2 – 4x – 21 1) Factors of -21: 1 -21, , ) Which pair adds to (- 4)? 3) Write the factors. x 2 – 4x – 21 = (x + 3)(x - 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 24: ) Which pair adds to -10 ? 3) Write the factors. x x + 24 = (x - 4)(x - 6) None of them adds to (-10). For the numbers to multiply to +24 and add to -10, they must both be negative!

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

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Solution #5: 2x 2 + x ) Multiply 2 (-21) = - 42; list factors of , , , , ) Which pair adds to 1 ? 3) Write the temporary factors. 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)in 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) Write the temporary factors. 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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