Question

Consider the polynomial expression.
((6x^3y^9) (7xy^2)) - ((9x^3y^2) (3xy^9)) + (8xy^2)^2
Select all the equivalent expressions.
1. 15x^4y^11 + 8x^2y^4
2. -15x^4y^11 + 64x^2y^4
3. 42x^4y^11 - 27x^4y^11 + 64x^2y^4
4. 15x^4y^11 + 64x^2y^4
5. 42x^3y^9 - 27x^4y^2 + 64x^22y^4

Answer 1

To simplify the given polynomial expression, we will go through each term step by step:

1. **First term: \((6x^3y^9)(7xy^2)\)**
- Multiply the coefficients: \(6 \times 7 = 42\)
- Add the exponents of \(x\): \(3 + 1 = 4\)
- Add the exponents of \(y\): \(9 + 2 = 11\)
- This term simplifies to \(42x^4y^{11}\).

2. **Second term: \((9x^3y^2)(3xy^9)\)**
- Multiply the coefficients: \(9 \times 3 = 27\)
- Add the exponents of \(x\): \(3 + 1 = 4\)
- Add the exponents of \(y\): \(2 + 9 = 11\)
- This term simplifies to \(27x^4y^{11}\).
- Since this term is subtracted, it will become \(-27x^4y^{11}\).

3. **Third term: \((8xy^2)^2\)**
- First, calculate \(8^2 = 64\).
- Multiply the coefficients: \(64\).
- Add the exponents of \(x\): \(1 \cdot 2 = 2\)
- Add the exponents of \(y\): \(2 \cdot 2 = 4\)
- This term simplifies to \(64x^2y^{4}\).

Now, we combine all three simplified terms together:
\[
42x^4y^{11} - 27x^4y^{11} + 64x^2y^{4}
\]
Combine the like terms for \(x^4y^{11}\):
\[
(42 - 27)x^4y^{11} + 64x^2y^{4} = 15x^4y^{11} + 64x^2y^{4}
\]

Thus, the overall simplified expression is:
\[
15x^4y^{11} + 64x^2y^{4}
\]

Next, we compare the equivalent expressions provided:

1. \(15x^4y^{11} + 8x^2y^{4}\) - **Not equivalent**
2. \(-15x^4y^{11} + 64x^2y^{4}\) - **Not equivalent**
3. \(42x^4y^{11} - 27x^4y^{11} + 64x^2y^{4}\) - **Equivalent**, as it simplifies back to the same expression.
4. \(15x^4y^{11} + 64x^2y^{4}\) - **Equivalent**
5. \(42x^3y^{9} - 27x^4y^{2} + 64x^{22}y^{4}\) - **Not equivalent**

Hence, the equivalent expressions are:
**3 and 4.**