Question

Explain the steps in multiplying the polynomials. Make sure you include the product in your answer. (x4+7)(5x2+10x−1)

Answer 1

To multiply the polynomials \((x^4 + 7)(5x^2 + 10x - 1)\), we will use the distributive property, often called the FOIL method for binomials, but it applies to polynomials of any degree as well. Here are the steps:

### Steps to Multiply the Polynomials:

1. **Distribute each term in the first polynomial to each term in the second polynomial**. This means you'll take each term in \(x^4 + 7\) and multiply it by each term in \(5x^2 + 10x - 1\).

2. **Multiply \(x^4\) by each term in the second polynomial**:
- Multiply \(x^4 \cdot 5x^2 = 5x^{6}\)
- Multiply \(x^4 \cdot 10x = 10x^{5}\)
- Multiply \(x^4 \cdot (-1) = -x^{4}\)

3. **Multiply \(7\) by each term in the second polynomial**:
- Multiply \(7 \cdot 5x^2 = 35x^{2}\)
- Multiply \(7 \cdot 10x = 70x\)
- Multiply \(7 \cdot (-1) = -7\)

4. **Combine all the results from the multiplications**:
- From \(x^4\): \(5x^6, 10x^5, -x^4\)
- From \(7\): \(35x^2, 70x, -7\)
- So, we combine these:
\[
5x^6 + 10x^5 - x^4 + 35x^2 + 70x - 7
\]

5. **Write the final result in standard form** (typically ordered by decreasing degree):
\[
5x^6 + 10x^5 - x^4 + 70x + 35x^2 - 7
\]

### Final Answer:
The product of the polynomials \((x^4 + 7)(5x^2 + 10x - 1)\) is:
\[
5x^6 + 10x^5 - x^4 + 35x^2 + 70x - 7
\]