do now pass out calculators. pick up a homework answer key from the back table and correct your...
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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.
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