SME
Grade 12 Math · Problem type
A repeated factor
Use this for “Write (3x − 1) ÷ (x − 2)² as partial fractions.”
Chart
Vertical asymptote x = 2
Horizontal asymptote y = 0
This is an example. Type your own values to start a new problem.
x
+
(
)
(x
−
)²
=
A
x
−
+
B
(x
−
)²
Formulas
A
=
a
3 = 3
B
=
a
×
p
+
b
5 = 3 × 2 + (−1)
Assumptions
•
A squared factor gets a fraction for each power: A ÷ (x − p) + B ÷ (x − p)².
•
Multiply through by (x − p)²: ax + b = A(x − p) + B.
•
Match the x terms for A, then put x = p for B.
Step-by-step
WE KNOW
a = 3, b = −1, p = 2
FIND
top over (x − p) (A), top over (x − p)² (B)
Step 1 · Find top over (x − p): A
Use A = a
The
x
terms
match:
A
(x
−
p
)
gives
Ax,
and
the
left
has
ax.
A
=
3
Step 2 · Find top over (x − p)² (B)
Use B = a × p + b
Put
x
=
p
into
ax
+
b
=
A
(x
−
p
)
+
B
:
the
A
term
is
0,
leaving
B
.
B
=
3
×
2
−
1
B
=
6
−
1
B
=
5
Check
Put the numbers back in. Both sides should match.
3 = 3 ✓
5 = 3 × 2 − 1 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.