Question

Explain the steps in multiplying the polynomials. Make sure you include the product in your answer.
(x^4 + 7) (5x^2 + 10x - 1)

Answer 1

To multiply the polynomials \((x^4 + 7)(5x^2 + 10x - 1)\), we can use the distributive property, commonly referred to as the FOIL method when dealing with two binomials, but in this case, we're multiplying a polynomial of degree 4 by a polynomial of degree 2.

Here are the steps to multiply these two polynomials:

1. **Distribute each term in the first polynomial by each term in the second polynomial**:
\[
(x^4 + 7)(5x^2 + 10x - 1) = x^4(5x^2) + x^4(10x) + x^4(-1) + 7(5x^2) + 7(10x) + 7(-1)
\]

2. **Perform the multiplications**:
- \(x^4(5x^2) = 5x^6\)
- \(x^4(10x) = 10x^5\)
- \(x^4(-1) = -x^4\)
- \(7(5x^2) = 35x^2\)
- \(7(10x) = 70x\)
- \(7(-1) = -7\)

3. **Compile all the resulting terms**:
Combine all the terms together:
\[
5x^6 + 10x^5 - x^4 + 35x^2 + 70x - 7
\]

4. **Arrange the terms in descending order**:
Organizing the terms by their degree, we get:
\[
5x^6 + 10x^5 - x^4 + 70x + 35x^2 - 7
\]

So, the final product of the multiplication of the polynomials \((x^4 + 7)(5x^2 + 10x - 1)\) is:
\[
\boxed{5x^6 + 10x^5 - x^4 + 35x^2 + 70x - 7}
\]