Cube Of A Binomial Worksheet - Fill Online, Printable, Fillable ... - Free Printable

To solve this matching worksheet, we need to expand each binomial expression using the Binomial Theorem or Pascal's Triangle.

The Binomial Theorem states that for any positive integer $n$:
$$(a + b)^n = \binom{n}{0}a^n b^0 + \binom{n}{1}a^{n-1}b^1 + \binom{n}{2}a^{n-2}b^2 + \dots + \binom{n}{n}a^0 b^n$$

Let's work through each problem one by one.

1. Expand $(3x + 5y)^3$
Here, $a = 3x$, $b = 5y$, and $n = 3$.
The coefficients for power 3 are 1, 3, 3, 1 (from Pascal's Triangle).
* Term 1: $1 \cdot (3x)^3 \cdot (5y)^0 = 1 \cdot 27x^3 \cdot 1 = 27x^3$
* Term 2: $3 \cdot (3x)^2 \cdot (5y)^1 = 3 \cdot 9x^2 \cdot 5y = 135x^2y$
* Term 3: $3 \cdot (3x)^1 \cdot (5y)^2 = 3 \cdot 3x \cdot 25y^2 = 225xy^2$
* Term 4: $1 \cdot (3x)^0 \cdot (5y)^3 = 1 \cdot 1 \cdot 125y^3 = 125y^3$

Result: $27x^3 + 135x^2y + 225xy^2 + 125y^3$
Looking at the options, this matches Option A.

2. Expand $(4x + 8y)^3$
Here, $a = 4x$, $b = 8y$, and $n = 3$.
Coefficients: 1, 3, 3, 1.
* Term 1: $1 \cdot (4x)^3 = 64x^3$
* Term 2: $3 \cdot (4x)^2 \cdot (8y) = 3 \cdot 16x^2 \cdot 8y = 384x^2y$
* Term 3: $3 \cdot (4x) \cdot (8y)^2 = 3 \cdot 4x \cdot 64y^2 = 768xy^2$
* Term 4: $1 \cdot (8y)^3 = 512y^3$

Result: $64x^3 + 384x^2y + 768xy^2 + 512y^3$
Looking at the options, this matches Option B.

3. Expand $(7x + 8y)^2$
Here, $a = 7x$, $b = 8y$, and $n = 2$.
Coefficients for power 2 are 1, 2, 1.
* Term 1: $1 \cdot (7x)^2 = 49x^2$
* Term 2: $2 \cdot (7x) \cdot (8y) = 112xy$
* Term 3: $1 \cdot (8y)^2 = 64y^2$

Result: $49x^2 + 112xy + 64y^2$
Looking at the options, this matches Option C.

4. Expand $(6x + 7y)^3$
Here, $a = 6x$, $b = 7y$, and $n = 3$.
Coefficients: 1, 3, 3, 1.
* Term 1: $1 \cdot (6x)^3 = 216x^3$
* Term 2: $3 \cdot (6x)^2 \cdot (7y) = 3 \cdot 36x^2 \cdot 7y = 756x^2y$
* Term 3: $3 \cdot (6x) \cdot (7y)^2 = 3 \cdot 6x \cdot 49y^2 = 882xy^2$
* Term 4: $1 \cdot (7y)^3 = 343y^3$

Result: $216x^3 + 756x^2y + 882xy^2 + 343y^3$
Looking at the options, this matches Option D.

5. Expand $(8x + 8y)^5$
Here, $a = 8x$, $b = 8y$, and $n = 5$.
This will result in very large numbers. Let's look at the first term to help identify the match.
First term: $\binom{5}{0}(8x)^5(8y)^0 = 1 \cdot 32768x^5 \cdot 1 = 32768x^5$.
Let's check the options for an expansion starting with $32768x^5$.
Option E starts with $32768x^5$. Let's verify the second term of Option E.
Second term calculation: $\binom{5}{1}(8x)^4(8y)^1 = 5 \cdot 4096x^4 \cdot 8y = 163840x^4y$.
Option E has $163840x^4y$. This looks correct.
Let's double check the last term: $(8y)^5 = 32768y^5$. Option E ends with $32768y^5$.

Result matches Option E.

6. Expand $(2x + 2y)^4$
Here, $a = 2x$, $b = 2y$, and $n = 4$.
Coefficients for power 4 are 1, 4, 6, 4, 1.
* Term 1: $1 \cdot (2x)^4 = 16x^4$
* Term 2: $4 \cdot (2x)^3 \cdot (2y) = 4 \cdot 8x^3 \cdot 2y = 64x^3y$
* Term 3: $6 \cdot (2x)^2 \cdot (2y)^2 = 6 \cdot 4x^2 \cdot 4y^2 = 96x^2y^2$
* Term 4: $4 \cdot (2x) \cdot (2y)^3 = 4 \cdot 2x \cdot 8y^3 = 64xy^3$
* Term 5: $1 \cdot (2y)^4 = 16y^4$

Result: $16x^4 + 64x^3y + 96x^2y^2 + 64xy^3 + 16y^4$
Looking at the options, this matches Option G.

7. Expand $(4x + 3y)^5$
Here, $a = 4x$, $b = 3y$, and $n = 5$.
Coefficients for power 5 are 1, 5, 10, 10, 5, 1.
* Term 1: $1 \cdot (4x)^5 = 1024x^5$
* Term 2: $5 \cdot (4x)^4 \cdot (3y) = 5 \cdot 256x^4 \cdot 3y = 3840x^4y$
* Term 3: $10 \cdot (4x)^3 \cdot (3y)^2 = 10 \cdot 64x^3 \cdot 9y^2 = 5760x^3y^2$
* Term 4: $10 \cdot (4x)^2 \cdot (3y)^3 = 10 \cdot 16x^2 \cdot 27y^3 = 4320x^2y^3$
* Term 5: $5 \cdot (4x) \cdot (3y)^4 = 5 \cdot 4x \cdot 81y^4 = 1620xy^4$
* Term 6: $1 \cdot (3y)^5 = 243y^5$

Result: $1024x^5 + 3840x^4y + 5760x^3y^2 + 4320x^2y^3 + 1620xy^4 + 243y^5$
Looking at the options, this matches Option F.

8. Expand $(6x + 8y)^3$
Here, $a = 6x$, $b = 8y$, and $n = 3$.
Coefficients: 1, 3, 3, 1.
* Term 1: $1 \cdot (6x)^3 = 216x^3$
* Term 2: $3 \cdot (6x)^2 \cdot (8y) = 3 \cdot 36x^2 \cdot 8y = 864x^2y$
* Term 3: $3 \cdot (6x) \cdot (8y)^2 = 3 \cdot 6x \cdot 64y^2 = 1152xy^2$
* Term 4: $1 \cdot (8y)^3 = 512y^3$

Result: $216x^3 + 864x^2y + 1152xy^2 + 512y^3$
Looking at the options, this matches Option H.

Final Answer:
1. A
2. B
3. C
4. D
5. E
6. G
7. F
8. H