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…

ax2 + bx+ c

EXAMPLE 1 Factor when b is negative and c is positive

Factor 2x2 – 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, 11, 2

–3x – 2x = –5x(x – 1)(2x – 3)–1, –31,2

Middle term

when multiplied

Possible

factorization

Factors

of 3

Factors of 2

Correct

2x2 – 7x + 3 = (x – 3)(2x – 1)ANSWER

EXAMPLE 2 Factor when b is positive and c is negative

Factor 3n2 + 14n – 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

3n2 + 14n – 5 = (n + 5)(3n – 1)ANSWER

GUIDED PRACTICE for Examples 1 and 2

Factor the trinomial.

1. 3t2 + 8t + 4 (t + 2)(3t + 2)ANSWER

2. 4s2 – 9s + 5 (s – 1)(4s – 5)ANSWER

3. 2h2 + 13h – 7 (h + 7)(2h – 1)ANSWER

SOLUTION

EXAMPLE 3 Factor when a is negative

Factor – 4x2 + 12x + 7.

STEP 1

Factor – 1 from each term of the trinomial.– 4x2 + 12x + 7 = – (4x2 – 12x – 7)

STEP 2

Factor the trinomial 4x2 – 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

– 4x2 + 12x + 7 = – (2x + 1)(2x – 7)

You can check your factorization using a graphing calculator. Graph y1 = –4x2 + 12x + 7 and y2 = (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. – 2y2 – 5y – 3 ANSWER – (y + 1)(2y + 3)

5. – 5m2 + 6m – 1 ANSWER – (m – 1)(5m – 1)

6. – 3x2 – x + 2 ANSWER – (x + 1)(3x – 2)

SOLUTION

EXAMPLE 3 Factor when a is negative

Factor – 4x2 + 12x + 7.

STEP 1

Factor – 1 from each term of the trinomial.– 4x2 + 12x + 7 = – (4x2 – 12x – 7)

STEP 2

Factor the trinomial 4x2 – 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

– 4x2 + 12x + 7 = – (2x + 1)(2x – 7)

You can check your factorization using a graphing calculator. Graph y1 = –4x2 + 12x + 7 and y2 = (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. – 2y2 – 5y – 3 ANSWER – (y + 1)(2y + 3)

5. – 5m2 + 6m – 1 ANSWER – (m – 1)(5m – 1)

6. – 3x2 – 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 = -16t2 + 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 = –16t2 + vt + s

h = –16t2 + 14t + 0

h = –16t2 + 14t

Vertical motion model

Substitute 14 for v and 0 for s.

Simplify.

EXAMPLE 4 Solve a multi-step problem

STEP 2Substitute 0 for h. When the armadillo lands, its height

above the ground is 0 feet. Solve for t.

0 = –16t2 + 14t

0 = 2t(–8t + 7)

2t = 0

t = 0

–8t + 7 = 0

t = 0.875

or

or Solve for t.

Zero-product property

Factor right side.

Substitute 0 for h.

ANSWER

The armadillo lands on the ground 0.875 second after the armadillo jumps.

EXAMPLE 5 Standardized Test Practice

w(3w + 13) = 10 Write an equation to model area.

3w2 + 13w2 – 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 =23

w Solve for w.

Reject the negative width.

ANSWERThe correct answer is A.

EXAMPLE 5 Guided Practice

12

mA 2 mC mB32 mD 3

2

ANSWER B

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: 6x2 – x – 2 with factoring x2 – x – 2

• Factor both of the problems above. • Write a few sentences explaining the

similarities and differences about the process of factoring each.