Do Now 2/19/19 Take out HW from last night. Copy HW in your planner. Punchline worksheet 3.15 Copy HW in your planner. Text p. 395, #4-32 multiples of 4 In your notebook, find the dimensions of the rectangle. The area is 44 square inches. (x – 5) (x – 12)

4 11 (x – 12) (x – 5) (16 – 12) (16 – 5) Problem Solving Find the dimensions of the rectangle. The area is 44 square inches. (x – 12) (x – 5) (16 – 12) 4 (16 – 5) 11

Homework Punchline worksheet 13.5 “What Happened to the Guy Who Lost the Pie-Eating Contest” He came in sickened

Learning Goal Learning Target SWBAT simplify, factor, and solve polynomial expressions and equations Learning Target SWBAT factor polynomials in the form ax² + bx + c

“Solving Equations By Factoring” Section 7.4 Remember this… “Solving Equations By Factoring” 2x² + 8x = 0 When using the zero-product property, sometimes you may need to factor the polynomial, or write it as a product of other polynomials. Look for the greatest common factor (GCF) of the polynomial’s terms. GCF- the monomial that divides evenly into EACH term of the polynomial. Look for common terms GCF

Section 7.5 “Factor x² + bx + c” Remember this… Section 7.5 “Factor x² + bx + c” Factoring x² + bx + c x² + bx + c = (x + p)(x + q) provided p + q = b and pq = c x² + 5x + 6 = (x + 3)(x + 2) Remember FOIL

Factoring polynomials Remember this… When factoring a trinomial, first consider the signs of p and q. (x + p)(x + q) x² + bx + c Signs of b and c (x + 2)(x + 3) x² + bx + c b is positive; c is positive b is negative; c is negative (x + 2)(x – 3) x² – bx – c b is positive; c is negative (x – 2)(x + 3) x² + bx – c (x – 2)(x – 3) x² – bx + c b is negative; c is positive

Factoring the polynomial. Then solve the equation. Remember this… Factoring the polynomial. Then solve the equation. x² - 7x - 30 Find two factors of -30 with different signs whose sum is -7. Factors of -30 Sum of factors -30, 1 -30 + 1 = -29 30, -1 30 + (-1) = 29 -2, 15 -2 + 15 = 13 2, -15 2 + (-15) = -13 -3, 10 -3 + 10 = 7 3, -10 3 + (-10) = -7 -5, 6 -5 + 6 = 1 5, -6 5 + (-6) = -1 (x + 3)(x – 10) x = -3, 10

Section 7.6 “Factor ax² + bx + c” 2x² – 7x + 3 First look at the signs of b and c. Factors of 2 Factors of 3 Possible factorization Middle term when multiplied 1, 2 1, 3 (x – 1)(2x – 3) -3x – 2x = -5x 3, 1 (x – 3)(2x – 1) -x – 6x = -7x (x – 3)(2x – 1)

Factor ax² + bx + c. Solve the equation. 3x² + 14x - 5 First look at the signs of b and c. Factors of 3 Factors of -5 Possible factorization Middle term when multiplied 1, 3 1, -5 (x + 1)(3x – 5) -5x + 3x = -2x 1,3 -1, 5 (x – 1)(3x + 5) 5x – 3x = 2x 5, -1 (x + 5)(3x – 1) -x + 15x = 14x -5, 1 (x – 5)(3x + 1) x – 15x = -14x (x +5)(3x – 1) x + 5 = 0 x = -5 3x – 1 = 0 x = 1/3

Factoring When ‘a’ is Negative First factor -1 from each term in the trinomial -4x² +12x + 7 Now look at the signs of b and c. – (4x² – 12x – 7) Factors of 4 Factors of -7 Possible factorization Middle term when multiplied 1, 4 1, -7 (x + 1)(4x – 7) -7x + 4x = -3x -1, 7 (x – 1)(4x + 7) 7x – 4x = 3x 7, -1 (x + 7)(4x – 1) -x + 28x = 27x -7, 1 (x – 7)(4x +1) x - 28x = -27x 2, 2 (2x + 1)(2x – 7) -14x +2x = -12x (2x – 1)(2x + 7) 14x – 2x = 12x Don’t forget about the negative you factored from the beginning!! – (2x + 1)(2x – 7) (2x + 1)(2x – 7)

Possible factorization 8x² – 14x – 15 (2x - 5)(4x + 3) Factors of 8 Factors of -15 Possible factorization Middle term 1, 8 1, -15 (x + 1)(8x – 15) 8x -15x = -7x -1, 15 (x – 1)(8x + 15) -8x + 15x = 7x 3, -5 (x + 3)(8x – 5) 24x -5x = 19x -3, 5 (x – 3)(8x +5) -24x +5x = -19x 5, -3 (x + 5)(8x – 3) 40x - 3x = 37x -5, 3 (x – 5)(8x + 3) -40x + 3x = -37x 15, -1 (x + 15)(8x – 1) 120x – 1x = 119x -15, 1 (x - 15)(8x + 1) -120x + 1x = -119x 2, 4 (2x + 1)(4x – 15) 4x -30x = -26x (2x – 1)(4x + 15) -4x + 30x = 26x (2x + 3)(4x – 5) 12x -10x = -2x (2x – 3)(4x +5) -12x +10x = -2x (2x + 5)(4x – 3) 20x - 6x = 14x (2x – 5)(4x + 3) -20x + 6x = -14x (2x + 15)(4x – 1) 60x – 2x = 58x (2x - 15)(4x + 1) -60x + 2x = -58x

A shortcut… for factoring ax² + bx + c (1) Split ‘a’ and ‘c’ terms. (2) multiply ‘a’ and ‘c’ together. (3) Then factor the product of ‘a’ and ‘c’ to equal the ‘b’ term. (4) Group terms into binomials. (5) Factor out GCF in both binomials. (6) Factor out common binomial.

Factor by grouping shortcut (1) Split ‘a’ and ‘c’ terms. (2) multiply ‘a’ and ‘c’ together. (3) Then factor the product of ‘a’ and ‘c’ to equal the ‘b’ term. (4) Group terms into binomials. (5) Factor out GCF in both binomials. (6) Factor out common binomial. 1 -120 2 -60 3 -40 4 -30 5 -24 6 -20 8x² – 14x – 15 Add to equal “b” 8x2 – 15 – 20x + 6x Group terms into binomials (8x2 – 20x) + (6x – 15) Factor GCF out each group 4x (2x – 5) + 3 (2x – 5) Factor out the common binomial (2x – 5) (4x + 3)

4t2 + 5 – 4t – 5t (4t2 – 4t) + (-5t + 5) Factor by grouping shortcut (1) Split ‘a’ and ‘c’ terms. (2) multiply ‘a’ and ‘c’ together. (3) Then factor the product of ‘a’ and ‘c’ to equal the ‘b’ term. (4) Group terms into binomials. (5) Factor out GCF in both binomials. (6) Factor out common binomial. -1 -12 -2 -10 -4 -5 4t² - 9t + 5 Add to equal “b” 4t2 + 5 – 4t – 5t Group terms into binomials (4t2 – 4t) + (-5t + 5) Factor GCF out each group 4t (t – 1) – 5 (t – 1) Factor out the common binomial (t – 1) (4t – 5 )

Factoring polynomials…Try it out… First look at the signs of b and c. 15t² + 40t + 20 Factor out GCF 5(3t² + 8t + 4) 5(t + 2)(3t + 2)

Try it out…Factoring When ‘a’ is Negative…Solve the Equation First factor -1 from each term in the trinomial -2x² +5x + 3 Now look at the signs of b and c. – (2x² – 5x – 3) Factors of 2 Factors of 3 Possible factorization Middle term when multiplied 1, 2 1, -3 (x + 1)(2x – 3) -3x + 2x = -x -1, 3 (x – 1)(2x + 3) 3x – 2x = x 3, -1 (x + 3)(2x – 1) -x + 6x = 5x -3, 1 (x – 3)(2x +1) x - 6x = -5x Don’t forget about the negative you factored from the beginning!! – (x – 3)(2x + 1) (x – 3)(2x + 1) x - 3 = 0 x = 3 2x + 1 = 0 x = -1/2

Homework Text p. 395, #4-32 multiples of 4