Do Now Pass out calculators. Pick up a homework answer key from the back table and correct your homework that was due on last week on Friday (pg. 586 # 4 – 40 even) * Use a different color pen/marker please

Do Now Pass out calculators. Pull out your Test Review Packet for Systems of Equations from last week and complete.

Objective: To factor polynomials in the form… ax 2 + bx+ c

EXAMPLE 1 Factor when b is negative and c is positive Factor 2x 2 – 7x + 3. SOLUTION Because b is negative and c is positive, both factors of c must be negative. Make a table to organize your work. You must consider the order of the factors of 3, because the x- terms of the possible factorizations are different.

EXAMPLE 1 Factor when b is negative and c is positive –x – 6x = –7x(x – 3)(2x – 1)  3,  1 1, 21, 2 –3x – 2x = –5x(x – 1)(2x – 3)–1, –31,2 Middle term when multiplied Possible factorization Factors of 3 Factors of 2 Correct 2x 2 – 7x + 3 = (x – 3)(2x – 1) ANSWER

EXAMPLE 2 Factor when b is positive and c is negative Factor 3n n – 5. SOLUTION Because b is positive and c is negative, the factors of c have different signs.

EXAMPLE 2 Factor when b is negative and c is positive n – 15n = –14n(n – 5)(3n + 1)–5, 11, 3 –n + 15n = 14n(n + 5)(3n – 1)5, –11, 3 5n – 3n = 2n(n – 1)(3n + 5)–1, 51, 3 –5n + 3n = – 2n(n + 1)(3n – 5)1, –51, 3 Middle term when multiplied Possible factorization Factors of –5 Factors of 3 Correct 3n n – 5 = (n + 5)(3n – 1) ANSWER

GUIDED PRACTICE for Examples 1 and 2 Factor the trinomial. 1. 3t 2 + 8t + 4(t + 2)(3t + 2) ANSWER 2. 4s 2 – 9s + 5(s – 1)(4s – 5) ANSWER 3. 2h h – 7 (h + 7)(2h – 1) ANSWER

SOLUTION EXAMPLE 3 Factor when a is negative Factor – 4x x + 7. STEP 1 Factor – 1 from each term of the trinomial. – 4x x + 7 = – (4x 2 – 12x – 7) STEP 2 Factor the trinomial 4x 2 – 12x – 7. Because b and c are both negative, the factors of c must have different signs. As in the previous examples, use a table to organize information about the factors of a and c.

EXAMPLE 3 Factor when a is negative 14x – 2x = 12x(2x – 1)(2x + 7)– 1, 72, 2 – 14x + 2x = – 12x(2x + 1)(2x – 7)1, – 72, 2 x – 28x = – 27x(x – 7)(4x + 1)– 7, 11, 4 7x – 4x = 3x(x – 1)(4x + 7)– 1, 71, 4 – x + 28x = 27x(x + 7)(4x – 1)7, – 11, 4 – 7x + 4x = – 3x(x + 1)(4x – 7)1, – 71, 4 Middle term when multiplied Possible factorization Factors of – 7 Factors of 4 Correct

EXAMPLE 3 Factor when a is negative ANSWER – 4x x + 7 = – (2x + 1)(2x – 7) You can check your factorization using a graphing calculator. Graph y 1 = –4x x + 7 and y 2 = (2x + 1)(2x – 7). Because the graphs coincide, you know that your factorization is correct. CHECK

GUIDED PRACTICE for Example 3 Factor the trinomial. 4. – 2y 2 – 5y – 3 ANSWER – (y + 1)(2y + 3) 5. – 5m 2 + 6m – 1 ANSWER – (m – 1)(5m – 1) 6. – 3x 2 – x + 2 ANSWER – (x + 1)(3x – 2)

SOLUTION EXAMPLE 3 Factor when a is negative Factor – 4x x + 7. STEP 1 Factor – 1 from each term of the trinomial. – 4x x + 7 = – (4x 2 – 12x – 7) STEP 2 Factor the trinomial 4x 2 – 12x – 7. Because b and c are both negative, the factors of c must have different signs. As in the previous examples, use a table to organize information about the factors of a and c.

EXAMPLE 3 Factor when a is negative 14x – 2x = 12x(2x – 1)(2x + 7)– 1, 72, 2 – 14x + 2x = – 12x(2x + 1)(2x – 7)1, – 72, 2 x – 28x = – 27x(x – 7)(4x + 1)– 7, 11, 4 7x – 4x = 3x(x – 1)(4x + 7)– 1, 71, 4 – x + 28x = 27x(x + 7)(4x – 1)7, – 11, 4 – 7x + 4x = – 3x(x + 1)(4x – 7)1, – 71, 4 Middle term when multiplied Possible factorization Factors of – 7 Factors of 4 Correct

EXAMPLE 3 Factor when a is negative ANSWER – 4x x + 7 = – (2x + 1)(2x – 7) You can check your factorization using a graphing calculator. Graph y 1 = –4x x + 7 and y 2 = (2x + 1)(2x – 7). Because the graphs coincide, you know that your factorization is correct. CHECK

GUIDED PRACTICE for Example 3 Factor the trinomial. 4. – 2y 2 – 5y – 3 ANSWER – (y + 1)(2y + 3) 5. – 5m 2 + 6m – 1 ANSWER – (m – 1)(5m – 1) 6. – 3x 2 – x + 2 ANSWER – (x + 1)(3x – 2)

Vertical Motion: A projectile is an object that is propelled into the air but has no power to keep itself in the air. A thrown ball is a projective, but an airplane is not. The height of a projectile can be described by the vertical motion model. The height h (in feet) of a projectile can be modeled by: h = -16t 2 + vt + s t = time (in seconds) the object has been in the air v = initial velocity (in feet per second) s = the initial height (in feet)

ARMADILLO PROBS.. EXAMPLE 4 Solve a multi-step problem A startled armadillo jumps straight into the air with an initial vertical velocity of 14 feet per second. After how many seconds does it land on the ground ?

SOLUTION EXAMPLE 4 Solve a multi-step problem STEP 1 Write a model for the armadillo’s height above the ground. h = –16t 2 + vt + s h = –16t t + 0 h = –16t t Vertical motion model Substitute 14 for v and 0 for s. Simplify.

EXAMPLE 4 Solve a multi-step problem STEP 2 Substitute 0 for h. When the armadillo lands, its height above the ground is 0 feet. Solve for t. 0 = –16t t 0 = 2t(–8t + 7) 2t = 0 t = 0 –8t + 7 = 0 t = or Solve for t. Zero-product property Factor right side. Substitute 0 for h. ANSWER The armadillo lands on the ground second after the armadillo jumps.

EXAMPLE 5 Standardized Test Practice w(3w + 13) = 10 Write an equation to model area. 3w w 2 – 10 = 0 Simplify and subtract 10 from each side. (w + 5)(3w – 2) = 0 Factor left side. w + 5 = 0 or 3w – 2 = 0 Zero-product property w = – 5 or = 2 3 w Solve for w. Reject the negative width. ANSWER The correct answer is A.

EXAMPLE 5 Guided Practice 1 2 m A 2 m C m B 3 2 m D 3 2 ANSWERB B A rectangle ’ s length is 1 inch more than twice its width. The area is 6 square inchs. What is the width? 9.

Exit Ticket: 1.Explain how you can use a graph to check a factorization. 2. Compare and contrast factoring: 6x 2 – x – 2 with factoring x 2 – x – 2 Factor both of the problems above. Write a few sentences explaining the similarities and differences about the process of factoring each.