SME
Grade 11 Math · Problem type
Add or subtract rational expressions
Use this for “Write 3/(x − 1) + 2/(x + 4) as one fraction.”
Chart
Zero: x = −2
Vertical asymptotes x = 1, x = −4
Horizontal asymptote y = 0
Zero of the result: −2
This is an example. Type your own values to start a new problem.
a
x
−
c
x
−
(
)
=
x
+
x²
+
x
+
(
)
Z
Zero of the result
worked out (not typed)
Formulas
A
=
a
+
t
×
c
5 = 3 + 1 × 2
B
=
−(
a
×
q
+
t
×
c
×
p
)
10 = −(3 × (−4) + 1 × 2 × 1)
S
=
−(
p
+
q
)
3 = −(1 + (−4))
P
=
p
×
q
−4 = 1 × (−4)
Z
=
−
B
A
−2
=
−
10
5
Assumptions
•
The common denominator is (x − p)(x − q) = x² + Sx + P.
•
Multiply each top by the other bottom: a(x − q) ± c(x − p) = Ax + B.
•
p and q are different, or the fractions already share a denominator.
Step-by-step
WE KNOW
a = 3, p = 1, t: +, c = 2, q = −4
FIND
x coefficient on top (A), constant on top (B), x coefficient below (S), constant below (P), zero of the result (Z)
Step 1 · Find x coefficient on top (A)
Use A = a + t × c
Over
the
common
bottom:
3(x + 4) + 2(x − 1)
(x − 1)(x + 4)
.
Its
x
terms,
with
t
=
−1
to
subtract.
A
=
3
+
1
×
2
A
=
3
+
2
A
=
5
Step 2 · Find constant on top (B)
Use B = −(a × q + t × c × p)
The
numbers:
a
×
(−
q
)
and
c
×
(−
p
),
combined
the
same
way.
B
=
−(3
×
(−4)
+
1
×
2
×
1)
B
=
−(−12
+
2)
B
=
−(−10)
B
=
10
Step 3 · Find x coefficient below (S)
Use S = −(p + q)
(x
−
p
)(x
−
q
)
has
x
terms
−px
−
qx.
S
=
−(1
−
4)
S
=
−(−3)
S
=
3
Step 4 · Find constant below (P)
Use P = p × q
(−
p
)
×
(−
q
)
is
pq.
P
=
1
×
(−4)
P
=
−4
Step 5 · Find zero of the result (Z)
Use Z = −B ÷ A
The
top
Ax
+
B
is
0
here:
the
graph
crosses
the
x-axis.
Z
=
−
10
5
3
x − 1
+
2
x + 4
=
5x + 10
x
2
+ 3x − 4
Z
=
−2
Check
Put the numbers back in. Both sides should match.
5 = 3 + 1 × 2 ✓
10 = −(3 × (−4) + 1 × 2 × 1) ✓
3 = −(1 − 4) ✓
−4 = 1 × (−4) ✓
−2
=
−
10
5
✓
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