Transcript Lesson 6-2

5-Minute Check on Chapter 2
Transparency 3-1
1. Evaluate 42 - |x - 7| if x = -3
2. Find 4.1  (-0.5)
Simplify each expression
4. (36d – 18) / (-9)
3. 8(-2c + 5) + 9c
5. A bag of lollipops has 10 red, 15 green, and 15 yellow lollipops.
If one is chosen at random, what is the probability that it is
not green?
6.
Standardized Test Practice:
Which of the following is a true
statement
A
8/4 < 4/8
B
-4/8 < -8/4
C
-4/8 > -8/4
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D
-4/8 > 4/8
Lesson 9-4
Factoring Trinomials:
ax2 + bx + c
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Objectives
• Factor trinomials of the form ax2 + bx + c
• Solve equations of the form ax2 + bx + c = 0
Vocabulary
• Prime polynomial– a polynomial that cannot be
written as a product of two polynomials with integral
coefficients
Working Backwards
• Start with the answer
• “Undo” the operation that got you to the
answer
• Keep “undoing” until you get back to the
beginning
Factoring ax2 + bx + c
• Set up table looking for factors m and n such
that mn = ac and m + n = b
Factors of ac
Sum of the factors
1, ac
ac + 1
• Group terms with common factors
• Factor the GCF from each grouping
• Rewrite as a product
Example 1a
Factor
In this trinomial,
and
You need to
find two numbers whose sum is 27 and whose product is
or 50. Make an organized list of factors of 50 and look
for the pair of factors whose sum is 27.
Factors of 50 Sum of Factors
1, 50
2, 25
51
27
The correct factors are 2 and 25.
Write the pattern
and
Example 1a cont
and
Group terms with
common factors
Factor the GCF
from each grouping
Answer:
Distributive Property
Check You can check this result by multiplying the two
factors.
F
O
I L
FOIL method
Simplify.
Example 1b
Factor
In this trinomial,
and
Since b is
negative,
is negative. Since c is positive, mn is
positive. So m and n must both be negative. Therefore,
make a list of the negative factors of
or 72, and look
for the pair of factors whose sum is –22.
Factors of 72 Sum of Factors
–1, –72
–2, –36
–4, –24
–4, –18
–73
–38
–27
–22
The correct factors are –4, –18.
Write the pattern
and
Example 1b cont
and
Group terms with
common factors
Factor the GCF
from each grouping
Answer:
Distributive Property
Example 2
Factor
Notice that the GCF of the terms
, and 32 is 4.
When the GCF of the terms of a trinomial is an integer
other than 1, you should first factor out this GCF.
Distributive Property
Now factor
Since the lead coefficient is 1, find
the two factors of 8 whose sum is 6.
Factors of 8 Sum of Factors
1, 8
2, 4
Answer: So,
complete factorization of
9
6
The correct factors are 2 and 4
Thus, the
is
Example 3
Factor
In this trinomial,
and
Since b is
positive,
is positive. Since c is negative, mn is
negative, so either m or n is negative, but not both.
Therefore, make a list of all the factors of 3(–5) or –15,
where one factor in each pair is negative. Look for the pair
of factors whose sum is 7.
Factors of –15
Sum of Factors
–1, 15
1, –15
–3, 5
3, –5
14
–14
2
–2
Answer:
There are no factors whose
sum is 7. Therefore,
cannot be
factored using integers.
is a prime polynomial.
Example 4
Solve
Original equation
Rewrite so one side equals 0.
Factor the left side.
or
Zero Product Property
Solve each equation.
Answer: The solution set is
Example 5
Model Rockets Ms. Nguyen’s science class built an
air-launched model rocket for a competition. When
they test-launched their rocket outside the classroom,
the rocket landed in a nearby tree. If the launch pad
was 2 feet above the ground, the initial velocity of the
rocket was 64 feet per second, and the rocket landed
30 feet above the ground, how long was the rocket in
flight? Use the equation
Vertical motion model
Example 5 cont
Subtract 30 from each side.
Factor out –4.
Divide each side by –4.
Factor
or
Zero Product Property
Solve each equation.
The solutions are 0.5 and 3.5 seconds. The first time represents
how long it takes the rocket to reach a height of 30 feet on its
way up. The second time represents how long it will take for the
rocket to reach the height of 30 feet again on its way down.
Thus the rocket will be in flight for 3.5 seconds before coming
down again.
Answer: 3.5 seconds
Summary & Homework
• Summary:
– Factoring as ax2 + bx + c: Find m and n whose
product is ac and whose sum is b. Then write as
ax2 + mx + nx + c and use factoring by grouping
• Homework:
– Pg. 499 24-28, 36-40, even