SME
Grade 12 Math · Precalculus
Partial fraction decomposition
Use this for “Write (5x + 1) ÷ ((x − 1)(x + 2)) as partial fractions.”
Chart
Vertical asymptotes x = 1, x = −2
Horizontal asymptote y = 0
This is an example. Type your own values to start a new problem.
x
+
(x
−
)(x
−
(
)
)
=
A
x
−
+
B
x
−
(
)
Formulas
A
=
a
×
p
+
b
p
−
q
2
=
5 × 1 + 1
1 − (−2)
B
=
a
×
q
+
b
q
−
p
3
=
5 × (−2) + 1
−2 − 1
a
=
A
+
B
5 = 2 + 3
Assumptions
•
(ax + b) ÷ ((x − p)(x − q)) = A ÷ (x − p) + B ÷ (x − q): one fraction for each linear factor.
•
Cover-up: multiply by (x − p) and put x = p, so the B term drops out and A is left.
•
Check by adding the two fractions back over the common bottom.
•
If the top’s degree is not less than the bottom’s, divide first and split the remainder.
Step-by-step
WE KNOW
a = 5, b = 1, p = 1, q = −2
FIND
top of the first fraction (A), top of the second fraction (B)
Step 1 · Find top of the first fraction (A)
Use A = (a × p + b) ÷ (p − q)
Cover
up
(x
−
p
)
on
the
left
and
put
x
=
p
into
what
is
left.
A
=
5 × 1 + 1
1 − (−2)
A
=
6
3
A
=
2
Step 2 · Find top of the second fraction (B)
Use B = (a × q + b) ÷ (q − p)
Cover
up
(x
−
q
)
and
put
x
=
q
into
what
is
left.
B
=
5 × (−2) + 1
−2 − 1
B
=
−10 + 1
−3
B
=
−
9
−3
B
=
3
Check
Put the numbers back in. Both sides should match.
2
=
5 × 1 + 1
1 − (−2)
✓
3
=
5 × (−2) + 1
−2 − 1
✓
5 = 2 + 3 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.