Cube of Binomial Worksheet: Complete with ease | airSlate SignNow - Free Printable

Problem Overview:

The task involves matching binomials to their expanded forms using the Binomial Theorem. The Binomial Theorem states that for any binomial \((a + b)^n\):

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]

Where:
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \).
- \( a \) and \( b \) are terms in the binomial.
- \( n \) is the power to which the binomial is raised.

We will expand each binomial given in the problem and match it with the corresponding expanded form provided in the options.

---

Step-by-Step Solution:

#### 1. Expand \((3x + 2y)^2\)
Using the Binomial Theorem for \( n = 2 \):
\[
(a + b)^2 = a^2 + 2ab + b^2
\]
Here, \( a = 3x \) and \( b = 2y \):
\[
(3x + 2y)^2 = (3x)^2 + 2(3x)(2y) + (2y)^2
\]
\[
= 9x^2 + 12xy + 4y^2
\]

Match: This matches option h: \( 9x^2 + 12xy + 4y^2 \).

---

#### 2. Expand \((4x + 3y)^2\)
Using the Binomial Theorem for \( n = 2 \):
\[
(a + b)^2 = a^2 + 2ab + b^2
\]
Here, \( a = 4x \) and \( b = 3y \):
\[
(4x + 3y)^2 = (4x)^2 + 2(4x)(3y) + (3y)^2
\]
\[
= 16x^2 + 24xy + 9y^2
\]

Match: This matches option g: \( 16x^2 + 24xy + 9y^2 \).

---

#### 3. Expand \((7x + 5y)^2\)
Using the Binomial Theorem for \( n = 2 \):
\[
(a + b)^2 = a^2 + 2ab + b^2
\]
Here, \( a = 7x \) and \( b = 5y \):
\[
(7x + 5y)^2 = (7x)^2 + 2(7x)(5y) + (5y)^2
\]
\[
= 49x^2 + 70xy + 25y^2
\]

Match: This matches option f: \( 49x^2 + 70xy + 25y^2 \).

---

#### 4. Expand \((8x + 7y)^2\)
Using the Binomial Theorem for \( n = 2 \):
\[
(a + b)^2 = a^2 + 2ab + b^2
\]
Here, \( a = 8x \) and \( b = 7y \):
\[
(8x + 7y)^2 = (8x)^2 + 2(8x)(7y) + (7y)^2
\]
\[
= 64x^2 + 112xy + 49y^2
\]

Match: This matches option e: \( 64x^2 + 112xy + 49y^2 \).

---

#### 5. Expand \((5x + 4y)^2\)
Using the Binomial Theorem for \( n = 2 \):
\[
(a + b)^2 = a^2 + 2ab + b^2
\]
Here, \( a = 5x \) and \( b = 4y \):
\[
(5x + 4y)^2 = (5x)^2 + 2(5x)(4y) + (4y)^2
\]
\[
= 25x^2 + 40xy + 16y^2
\]

Match: This matches option d: \( 25x^2 + 40xy + 16y^2 \).

---

#### 6. Expand \((3x + 2y)^4\)
Using the Binomial Theorem for \( n = 4 \):
\[
(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4
\]
Here, \( a = 3x \) and \( b = 2y \):
\[
(3x + 2y)^4 = (3x)^4 + 4(3x)^3(2y) + 6(3x)^2(2y)^2 + 4(3x)(2y)^3 + (2y)^4
\]
\[
= 81x^4 + 4(27x^3)(2y) + 6(9x^2)(4y^2) + 4(3x)(8y^3) + 16y^4
\]
\[
= 81x^4 + 216x^3y + 216x^2y^2 + 96xy^3 + 16y^4
\]

Match: This matches option c: \( 81x^4 + 216x^3y + 216x^2y^2 + 96xy^3 + 16y^4 \).

---

#### 7. Expand \((4x + 3y)^3\)
Using the Binomial Theorem for \( n = 3 \):
\[
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
\]
Here, \( a = 4x \) and \( b = 3y \):
\[
(4x + 3y)^3 = (4x)^3 + 3(4x)^2(3y) + 3(4x)(3y)^2 + (3y)^3
\]
\[
= 64x^3 + 3(16x^2)(3y) + 3(4x)(9y^2) + 27y^3
\]
\[
= 64x^3 + 144x^2y + 108xy^2 + 27y^3
\]

Match: This matches option b: \( 64x^3 + 144x^2y + 108xy^2 + 27y^3 \).

---

#### 8. Expand \((8x + 9y)^2\)
Using the Binomial Theorem for \( n = 2 \):
\[
(a + b)^2 = a^2 + 2ab + b^2
\]
Here, \( a = 8x \) and \( b = 9y \):
\[
(8x + 9y)^2 = (8x)^2 + 2(8x)(9y) + (9y)^2
\]
\[
= 64x^2 + 144xy + 81y^2
\]

Match: This matches option a: \( 64x^2 + 144xy + 81y^2 \).

---

Final Matches:

1. \((3x + 2y)^2\) → h: \( 9x^2 + 12xy + 4y^2 \)
2. \((4x + 3y)^2\) → g: \( 16x^2 + 24xy + 9y^2 \)
3. \((7x + 5y)^2\) → f: \( 49x^2 + 70xy + 25y^2 \)
4. \((8x + 7y)^2\) → e: \( 64x^2 + 112xy + 49y^2 \)
5. \((5x + 4y)^2\) → d: \( 25x^2 + 40xy + 16y^2 \)
6. \((3x + 2y)^4\) → c: \( 81x^4 + 216x^3y + 216x^2y^2 + 96xy^3 + 16y^4 \)
7. \((4x + 3y)^3\) → b: \( 64x^3 + 144x^2y + 108xy^2 + 27y^3 \)
8. \((8x + 9y)^2\) → a: \( 64x^2 + 144xy + 81y^2 \)

Final Answer:

\[
\boxed{
\begin{aligned}
1 & : h \\
2 & : g \\
3 & : f \\
4 & : e \\
5 & : d \\
6 & : c \\
7 & : b \\
8 & : a \\
\end{aligned}
}
\]