SME
Grade 11 Math · Problem type
Expand (ax + b)ⁿ
Use this for “Expand (x + 2)⁴.”
Diagram
Row 4: 1, 4, 6, 4, 1; they add to 2⁴ = 16.
(A + B)⁴ = A⁴ + 4A³B + 6A²B² + 4AB³ + B⁴
This is an example. Type your own values to start a new problem.
(
x
+
)
c₄
Coefficient of x⁴
worked out (not typed)
c₃
Coefficient of x³
worked out (not typed)
c₂
Coefficient of x²
worked out (not typed)
c₁
Coefficient of x
worked out (not typed)
c₀
Constant term
worked out (not typed)
Formulas
coefficient
of
x
6
=
C(
n
,
6)
×
a
6
×
b
n
− 6
coefficient
of
x
6
=
C(4,
6)
×
1
6
×
2
4 − 6
coefficient
of
x
5
=
C(
n
,
5)
×
a
5
×
b
n
− 5
coefficient
of
x
5
=
C(4,
5)
×
1
5
×
2
4 − 5
coefficient
of
x
4
=
C(
n
,
4)
×
a
4
×
b
n
− 4
coefficient
of
x
4
=
C(4,
4)
×
1
4
×
2
4 − 4
coefficient
of
x
3
=
C(
n
,
3)
×
a
3
×
b
n
− 3
coefficient
of
x
3
=
C(4,
3)
×
1
3
×
2
4 − 3
coefficient
of
x
2
=
C(
n
,
2)
×
a
2
×
b
n
− 2
coefficient
of
x
2
=
C(4,
2)
×
1
2
×
2
4 − 2
coefficient
of
x
=
C(
n
,
1)
×
a
1
×
b
n
− 1
coefficient
of
x
=
C(4,
1)
×
1
1
×
2
4 − 1
c₀
=
C(
n
,
0)
×
a
0
×
b
n
− 0
16
=
C(4,
0)
×
1
0
×
2
4 − 0
Assumptions
•
Row n of Pascal’s triangle gives the coefficients: 1, 4, 6, 4, 1 for n = 4.
•
The xʲ term is C(n, j) × aʲ × bⁿ⁻ʲ: the powers of a climb as the powers of b fall.
•
There are n + 1 terms; a power of x above n has coefficient 0.
Step-by-step
WE KNOW
a = 1, b = 2, n = 4
FIND
coefficient of x⁴ (c₄), coefficient of x³ (c₃), coefficient of x² (c₂), coefficient of x (c₁), constant term (c₀)
Step 1 · Find coefficient of x⁴ (c₄)
Use coefficient of x⁴ = C(n, 4) × a⁴ × bⁿ⁻⁴
Row
n
,
entry
4,
times
a
to
the
4
and
b
to
the
rest
of
the
power.
c₄
=
C(
n
,
4)
×
a
4
×
b
n
− 4
c₄
=
C(4,
4)
×
1
4
×
2
4 − 4
c₄
=
1
×
1
4
×
2
0
c₄
=
1
×
1
×
1
c₄
=
1
Step 2 · Find coefficient of x³ (c₃)
Use coefficient of x³ = C(n, 3) × a³ × bⁿ⁻³
Row
n
,
entry
3,
times
a
to
the
3
and
b
to
the
rest
of
the
power.
c₃
=
C(
n
,
3)
×
a
3
×
b
n
− 3
c₃
=
C(4,
3)
×
1
3
×
2
4 − 3
c₃
=
4
×
1
3
×
2
1
c₃
=
4
×
1
×
2
c₃
=
8
Step 3 · Find coefficient of x² (c₂)
Use coefficient of x² = C(n, 2) × a² × bⁿ⁻²
Row
n
,
entry
2,
times
a
to
the
2
and
b
to
the
rest
of
the
power.
c₂
=
C(
n
,
2)
×
a
2
×
b
n
− 2
c₂
=
C(4,
2)
×
1
2
×
2
4 − 2
c₂
=
6
×
1
2
×
2
2
c₂
=
6
×
1
×
4
c₂
=
24
Step 4 · Find coefficient of x (c₁)
Use coefficient of x = C(n, 1) × a¹ × bⁿ⁻¹
Row
n
,
entry
1,
times
a
to
the
1
and
b
to
the
rest
of
the
power.
c₁
=
C(
n
,
1)
×
a
1
×
b
n
− 1
c₁
=
C(4,
1)
×
1
1
×
2
4 − 1
c₁
=
4
×
1
1
×
2
3
c₁
=
4
×
1
×
8
c₁
=
32
Step 5 · Find constant term (c₀)
Use c₀ = C(n, 0) × a⁰ × bⁿ⁻⁰
Row
n
,
entry
0,
times
a
to
the
0
and
b
to
the
rest
of
the
power.
c₀
=
C(4,
0)
×
1
0
×
2
4 − 0
c₀
=
1
×
1
0
×
2
4
c₀
=
1
×
1
×
16
(x + 2)
4
=
x
4
+
8x³
+
24x²
+
32x
+
16
c₀
=
16
Check
Put the numbers back in. Both sides should match.
no
x
6
term:
the
powers
stop
at
n
✓
no
x
5
term:
the
powers
stop
at
n
✓
1
=
1
×
1
4
×
2
0
✓
8
=
4
×
1
3
×
2
1
✓
24
=
6
×
1
2
×
2
2
✓
32
=
4
×
1
1
×
2
3
✓
16
=
1
×
1
0
×
2
4
✓
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Pascal’s rule: add the two above
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