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Factoring Trinomials
THE X-BOX
METHOD
of the form
ax2 + bx + c
where a ≠ 1 and the GCF of a, b, c = 1
b=m+n
GCF of
ax2 and nx
KFHS
Algebra 2
GCF of
mx and c
b
m
n
ac
ac = m ∙ n
GCF of
ax2 and mx
ax2
mx
GCF of
nx and c
nx
c
Example: Factor 3x2 + 11x – 20
Step 1: Identify a, b, c and fill in b and ac in the X.
a=3
Step 3: Draw your box and fill in ax2 , c, mx, nx.
15 + (-4) = 11
11
11
15
b = 11
c = -20
Step 2: Identify m & n such that their product is
ac and their sum is b. In other words, m & n are
factors of ac that add up to b.
-60
11
15
-4
3x2
15x
-4x
-20
-4
-60
-60
15 ∙ (-4) = -60
Step 4: Find the GCF of each column and write it
above the appropriate column. GCF of ax2 and nx,
GCF of mx and c.
x
5
3x2
15x
-4x
-20
Step 5: Find the GCF of each row and write it to
the left of the appropriate row. GCF of ax2 and mx,
GCF of nx and c.
3x
-4
x
5
3x2
15x
-4x
Step 6: Write the factors from the top and left of
the box in parentheses.
x
5
3x
3x2
15x
-4
-4x
-20
(x + 5) (3x – 4)
-20
Step 7: Multiply using FOIL to confirm the factors
are correct.
(x + 5) (3x – 4)
3x2 – 4x + 15x – 20
3x2 + 11x - 20
THE X-BOX
METHOD
KFHS
Algebra 2
Factoring Trinomials
The Slip & Slide
Method
of the form
ax2 + bx + c
where a ≠ 1 and the GCF of a, b, c = 1
1)
Multiply c by a (SLIP the a to the end)
2)
Rewrite problem with the new c and with a set to equal 1.
3)
Find factors of the new c that add up to b.
4)
Write the factors as a product.
5)
SLIDE the original a back in (divide by a)
6)
Reduce. If the fraction reduces evenly, do so.
If the fraction does not reduce evenly, put the denominator in front of the x and
leave the numerator where it is.
7)
Use FOIL to check your factors; make sure that they multiply back to the original
quadratic expression.
Example: Factor 3x2 + 11x – 20
Step 1: Identify a, b, c and multiply a and c.
SLIP the a to the end by multiplying.
Step 2: Rewrite problem with new c and a set to
1
a = 3 b = 11
a*c = -60
x2 + 11x – 60
c = -20
Step 3: Find factors of the new c that add up to b
The numbers that multiply to get -60
and add to get 11 are -4 and 15.
On calculator, go to y= and put in -60/x. Hit 2nd
Graph and look at the table to find the factors of
-60. Find the two that add up to 11.
Step 4: Write the factors as a product.
(x – 4) (x + 15)
Step 6: REDUCE
If fractions reduce evenly, do so.
If not, move denominator in front of the x, and
leave the numerator.
(𝟑𝒙 − 𝟒)(𝒙 + 𝟓)
Step 5: SLIDE the 6 (the original value of a) back
in by dividing.
𝟒
𝟏𝟓
(𝒙 − )(𝒙 + )
𝟑
𝟑
Step 6: REDUCE
𝟒
(𝒙 − )(𝒙 + 𝟓)
𝟑
Step 7: Multiply using FOIL to confirm the factors
are correct.
(3x - 4) (x + 5)
3x2 + 15x - 4x – 20
3x2 + 11x - 20
The Slip & Slide
Method
Factoring Trinomials
Completing the Square
Grouping
Method
Method
of the form
ax2 + bx
=c
Factoring
Trinomials
where a = 1 andofthe
of a, b, c = 1
theGCF
form
ax2 + bx + c
where the GCF of a, b, & c = 1
1)
Just like when using the x-box or slip-and-slide methods, find the factors of ac
that add up to b.
2)
Rewrite the middle term as the sum of the two factors of ac. Group the factors
such that each one has a common factor with either a or c.
3)
Pull out the common factor(s) of each group.
4)
If this method can be used, you will end up with the same thing inside both
parentheses. This is one of your factors.
5)
The factors that you pulled out of each group form the other factor.
6)
Use FOIL to check your factors; make sure that they multiply back to the original
quadratic expression.
Example: Factor 3x2 + 11x – 20
Step 1 Find factors of the new c that add up to b
The numbers that multiply to get -60 and add to
get 11 are -4 and 15.
On calculator, go to y= and put in -60/x. Hit 2nd
Graph and look at the table to find the factors of
-60. Find the two that add up to 11.
Step 2 Rewrite the middle term as the sum of the Step 3: Pull out the common factor(s) of each
two factors of ac. Group the factors such that
group.
each one has a common factor with either a or c.
(3x2 + 15x) – (4x – 20)
3x(x+5) – 4(x+5)
Step 4: Pull out the common factor(s) of each
group.
We put the 15 with the 3 because they share the
factor of 3
We put the -4 with the -20 because they share
the factor of -4.
Step 5: The factors that you pulled out of each
group form the other factor.
Step 6: Multiply using FOIL to confirm the factors
are correct.
3x(x+5) – 4(x+5)
3x(x+5) – 4(x+5)
(3x - 4) (x + 5)
Since we have the same (x+5) inside both
parentheses, this is one of our factors.
Since we pulled a 3x out of one group and a -4
out of the other, they form the second factor,
3x-4.
3x2 + 15x - 4x – 20
3x2 + 11x - 20
(3x-4)(x+5)
Grouping
Method