SME
Grade 9 Math · Problem type
Factor when a is not 1
Use this for “Factor 2x² + 7x + 3.”
Diagram
The tiles of 2x² + 7x + 3 make a rectangle (2x + 1) by (x + 3): (2x + 1)(x + 3) = 2x² + 7x + 3.
This is an example. Type your own values to start a new problem.
a
x²
+
b
x
+
c
=
(
p
x
+
q
)(
r
x
+
s
)
A coefficient of 1 is not written: 1x is written x.
m
First part of b
worked out (not typed)
n
Second part of b
worked out (not typed)
Formulas
m
=
the
number
in
the
pair
of
a
×
c
that
adds
to
b
6 = the number in the pair of 2 × 3 that adds to 7
n
=
b
−
m
1 = 7 − 6
p
=
greatest
common
factor
of
a
and
m
2 = greatest common factor of 2 and 6
r
=
a
p
1
=
2
2
s
=
m
p
3
=
6
2
q
=
n
r
1
=
1
1
Assumptions
•
Take out any common factor of a, b and c first, or a bracket keeps one.
•
Find two numbers m and n that multiply to ac and add to b.
•
Split bx into mx + nx, then factor each pair: both leave the same bracket.
•
Check by multiplying the brackets back out.
Step-by-step
WE KNOW
a = 2, b = 7, c = 3
FIND
first part of b (m), second part of b (n), x in the first factor (p), x in the second factor (r), number in the second factor (s), number in the first factor (q)
Step 1 · Find first part of b (m)
Use m = the number in the pair of a × c that adds to b
Multiply
a
by
c
,
then
find
the
factor
pair
of
ac
that
adds
to
b
.
m
=
the
number
in
the
pair
of
2
×
3
that
adds
to
7
2 × 3 = 6
6 × 1 = 6
6 + 1 = 7
m
=
6
Step 2 · Find second part of b (n)
Use n = b − m
The
two
parts
add
to
b
.
n
=
7
−
6
n
=
1
→
2x²
+
6x
+
x
+
3
Step 3 · Find x in the first factor (p)
Use p = greatest common factor of a and m
Take the greatest common factor out of the first pair, ax² + mx.
p
=
greatest
common
factor
of
2
and
6
p
=
2
Step 4 · Find x in the second factor (r)
Use r = a ÷ p
What is left of ax² in the bracket, after px comes out.
r
=
2
2
r
=
1
Step 5 · Find number in the second factor (s)
Use s = m ÷ p
What
is
left
of
mx
in
the
bracket:
the
common
bracket
is
(rx
+
s
).
s
=
6
2
s
=
3
Step 6 · Find number in the first factor (q)
Use q = n ÷ r
The
second
pair,
nx
+
c
,
is
q
times
the
same
bracket.
q
=
1
1
q
=
1
→
2x(x
+
3)
+
1(x
+
3)
=
(2x
+
1)(x
+
3)
Check
Put the numbers back in. Both sides should match.
6 × 1 = 6 ✓
6 + 1 = 7 ✓
2 = greatest common factor of 2 and 6 ✓
1
=
2
2
✓
3
=
6
2
✓
1
=
1
1
✓
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Factoring polynomials: GCF, trinomials and special products
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Take out the greatest common factor
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