73 Factoring Trinomials

Factoring Trinomials Multiplying Binomials (FOIL) Multiply (x+3)(x+2) Distribute x•x+x•2+3•x+3•2 F O I L = x2+ 2x + 3x + = x2+ 5x + Multiplying Binomials (Tiles) Multiply (x+3)(x+2) Using Algebra Tiles, we have: x + x x2 x x x = x2 + 5x + + x 1 x 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 twelve “1” tiles x 1 1 1 2) Add seven “x” tiles 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 twelve “1” tiles x 1 1 1 2) Add seven “x” tiles 3) Rearrange the tiles until they form a rectangle! 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 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 x (vertical or horizontal, at least one of each) and x 1 1 twelve “1” tiles x 1 1 3) Rearrange the tiles until they form a rectangle! x 1 1 2) Add seven “x” tiles 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: 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 + x x2 x x x x + x 1 1 x 1 1 x 1 1 x2 + 7x + 12 = ( x + 4)( x + 3) Factoring Trinomials (Method 2) Again, we will factor trinomials such as x2 + 7x + 12 back into binomials This method does not use tiles, instead we look for the pattern of products and sums! 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 = • 12 =2•6 =3•4 Factoring Trinomials (Method 2) x2 + 7x + 12 Step 2: Choose the pair that adds up to the middle coefficient 12 = • 12 =2•6 =3•4 Step 3: Fill those numbers into the blanks in the binomials: ( x + )( x + ) x2 + 7x + 12 = ( x + 3)( x + 4) Factoring Trinomials (Method 2) 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 = • -24, -1 • 24 = • -12, -2 • 12 = • -8, -3 • = • -6, - • Step 2: Which pair adds up to 2? Step 3: Write the binomial factors x2 + 2x - 24 = ( x - 4)( x + 6) Factoring Trinomials (Method 2*) Factor 3x2 + 14x + This time, the x2 term DOES have a coefficient (other than 1)! Step 1: Multiply • = 24 (the leading coefficient & constant) 24 = • 24 = • 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 (Method 2*) Factor 3x2 + 14x + Step 4: Write temporary factors with the two numbers ( x + )( x + 12 ) 3 Step 5: Put the original leading coefficient (3) under both numbers ( x + )( x + 12 ) 3 Step 6: Reduce the fractions, if possible ( x + )( x + ) Step 7: Move denominators in front of x ( 3x + )( x + ) Factoring Trinomials (Method 2*) Factor 3x2 + 14x + You should always check the factors by distributing, especially since this process has more than a couple of steps ( 3x + )( x + ) = 3x • x + 3x • + • x + • = 3x2 + 14 x + √ 3x2 + 14x + = (3x + 2)(x + 4) Factoring Trinomials (Method 2*) Factor 3x2 + 11x + This time, the x2 term DOES have a coefficient (other than 1)! Step 1: Multiply • = 12 12 = • 12 (the leading coefficient & constant) Step 2: List all pairs of numbers that multiply to equal that product, 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 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 • -21, -1 • 21 • -7, -3 • 2) Which pair adds to (- 4)? 3) Write the factors t2 – 4t – 21 = (t + 3)(t - 7) Solution #2: x2 + 12x + 32 • 32 • 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: • 24 • 12 3•8 4•6 1) Factors of 32: 2) Which pair adds to -10 ? x2 - 10x + 24 -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 • -18, -1 • 18 • -9, -2 • • -6, -3 • 2) Which pair adds to ? 3) Write the factors x2 + 3x - 18 = (x - 3)(x + 18) Solution #5: 1) Multiply • (-21) = - 42; list factors of - 42 2) Which pair adds to ? 3) Write the temporary factors 4) Put “2” underneath 2x2 + x - 21 • -42, -1 • 42 • -21, -2 • 21 • -14, -3 • 14 • -7, -6 • ( x - 6)( x + 7) 2 5) Reduce (if possible) ( x - 6)( x + 7) 2 6) Move denominator(s)in front of “x” ( x - 3)( 2x + 7) 2x2 + x - 21 = (x - 3)(2x + 7) Solution #6: 1) Multiply • 10 = 30; list factors of 30 2) Which pair adds to 11 ? 3) Write the temporary factors 4) Put “3” underneath 3x2 + 11x + 10 • 30 • 15 • 10 5•6 ( x + 5)( x + 6) 3 5) Reduce (if possible) ( x + 5)( x + 6) 3 6) Move denominator(s)in front of “x” ( 3x + 5)( x + 2) 3x2 + 11x + 10 = (3x + 5)(x + 2) ... (x+3)(x+2) Using Algebra Tiles, we have: x + x x2 x x x = x2 + 5x + + x 1 x 1 Factoring Trinomials (Tiles) How can we factor trinomials such as x2 + 7x + 12 back into binomials? One method is to again... least one of each) and x 1 1 twelve “1” tiles x 1 1 1 2) Add seven “x” tiles Factoring Trinomials (Tiles) How can we factor trinomials such as x2 + 7x + 12 back into binomials? One method is to again... 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