Download X-Puzzles X- Box X- Box X-Puzzles Factor the x

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X-Puzzles
Chapter 8
Using the pattern in puzzles A and B, complete puzzles C, D and E.
Section 3 day 1
Do you get it?
Factoring Trinomials with a lead coefficient of
ONE
A.
3
With the BOXes
6
5
B.
4
2
12
7
C.
3
10
5 2
7
D.
21
3 7
10
E.
24
12
2
14
Did you discovered the pattern?
X- Box
X- Box
Product
3
Product
(3)(-9)
-9
3
3
-9
-27
-9
3
-9
Sum
Sum
3 + (-9)
-6
Factor the x-box way
X-Puzzles
Example: Factor 3x2 -13x -10
Try these
(3)(-10)=
-30
A.
B.
8
4
2
6
C.
-6
3
2
1
D.
-15
5
3
2
E.
18
6
3
-9
16x2
8x
2x
10x
2
-15
x
-5
3x
3x2
-15x
+2
2x
-10
-13
Did you have the pattern?
3x2 -13x -10 = (x-5)(3x+2)
1
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Examples
Factor the x-box way
Factor using the x-box and the BOX methods.
1. x2 + 4x – 12
y = ax2 + bx + c
Product
First and Last
Coefficients
ac=mn
n
m
b=m+n
Sum
Middle
Base 1
Base 2
1st
Term
Factor
n
GCF
Factor
m
Height
a)
-4
-5
-2
4
+6
x
6x
-2
-2x
-12
One more…
 Given: x2 - 10x - 24
x
b)
x
-5
-9
x
x2
Solution: x2 + 4x – 12 = (x + 6)(x - 2)
2. x2 - 9x + 20
20
6
Last
term
Examples continued
a)
b)
-12
-4
x2 -4x
-5x
20
Solution: x2 - 9x + 20 = (x - 4)(x - 5)
-24x2
x
-12x
x
2
x2
2x
-12x
-24
2x
-10x
-12
Same numbers that
were in the x-puzzle!
Answer: (x + 2)(x - 12)
Chapter 8
Other methods
Section 3
Factoring Trinomials with a lead coefficient of ONE
Solving equation with trinomials
For comparison and development later
If needed from yesterday other wise skip
go TO THE NEXT SLIDE WITH THE LARGE
ORANGE BAND
2
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This one is more mental math and works
because the coefficient on the x2 is 1
Factor
x2
7 x 12
In this trinomial, b = 7 and c = 12. You need to find the
two numbers whose sum is 7 and whose product is 12.
Make an organized list of the factors of 12, and look for
the pair of factors whose sum is 7.
Check You can check the result by multiplying the
two factors.
F
O
I
L
FOIL method
Simplify.
Factors of 12 Sum of Factors
1, 12
2, 6
3, 4
Or check with the BOX METHOD
13
8
7
x
+4
The correct factors are
3 and 4.
Write the pattern.
Answer:
x
x2
4x
3
3x
16
and
This one is more mental math and works
because the coefficient on the x2 is 1
Factor
In this trinomial,
and
This means
is
negative and mn is positive. So m and n must both be
negative. Therefore, make a list of the negative factors of
27, and look for the pair whose sum is –12.
Factors of 27 Sum of Factors
–28
–1, –27
–12
–3, –9
Answer:
The correct factors are
–3 and –9.
Write the pattern.
and
Factor
In this trinomial,
and
This means
is
positive and mn is negative, so either m or n is negative,
but not both. Therefore, make a list of the factors of –18
where one factor of each pair is negative. Look for the pair
of factors whose sum is 3.
Factors of –18
1, –18
–1, 18
2, –9
–2, 9
3, –6
–3, 6
Check You can check this result by using a graphing
calculator. Graph
and
on the same screen. Since only one graph appears,
the two graphs must coincide. Therefore, the trinomial
has been factored correctly.
Write the pattern.
Answer:
and
Sum of Factors
–17
17
– 7
7
The correct factors
– 3
are –3 and 6.
3
3
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Factor
Since
and
is negative and mn is
negative. So either m or n is negative, but not both.
Factors of –20
Sum of Factors
1, –20
–1, 20
2, –10
–2, 10
4, –5
–4, 5
–19
19
– 8
8
– 1
1
Answer:
Write the pattern.
and
The correct factors are
4 and –5.
A.)
Factor
B.)
Answer:
Factor
Answer:
Solving trinomials Equations
Using the Zero Property
C.)
D.)
Factor
Factor
Answer:
Answer:
Solve
Check your solutions.
Check Substitute –5 and 3 for x in the original equation.
Original equation
I did not show the X box due to space
Rewrite the equation so that
one side equals 0.
Factor.
or
Zero Product Property
Solve each equation.
Answer: The solution is
4
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Solve
Check your solutions.
1.) Set equal to zero
2.) Factor.
3.) Use the zero property
4.) check
Architecture Marion has a small art studio measuring
10 feet by 12 feet in her backyard. She wants to build a
new studio that has three times the area of the old
studio by increasing the length and width by the same
amount. What will be the dimensions of the new
studio?
Explore Begin by
making a diagram like the
one shown to the right,
labeling the appropriate
dimensions.
Answer:
Plan
Let
the amount added to each dimension of
the studio.
The new length times the new width equals the new area.
Factor.
or
Zero Product
Property
Solve each equation.
old area
Solve
Examine The solution set is
Only 8 is a valid
solution, since dimensions cannot be negative.
Write the equation.
Multiply.
Subtract 360 from
each side.
Answer: The length of the new studio should be
or 20 feet and the new width should be
or 18 feet.
Photography Adina has a
photograph. She wants
to enlarge the photograph by increasing the length
and width by the same amount. What dimensions of
the enlarged photograph will be twice the area of the
original photograph?
Answer:
5