Questions: 3b^4(8-4b^5)

Solution Steps

To solve the problem of multiplying a univariate polynomial by a monomial, we will use the distributive property. This involves multiplying the monomial by each term inside the parentheses separately and then combining the results.

The distributive property states that for any numbers \(a\), \(b\), and \(c\), the expression \(a(b + c)\) can be expanded to \(ab + ac\). In this problem, we need to apply the distributive property to the expression:

\[ 3b^4(8 - 4b^5) \]

This means we will multiply \(3b^4\) by each term inside the parentheses.

1. Multiply \(3b^4\) by \(8\): \[ 3b^4 \times 8 = 24b^4 \]

2. Multiply \(3b^4\) by \(-4b^5\): \[ 3b^4 \times (-4b^5) = -12b^{4+5} = -12b^9 \]

Combine the results from the multiplication:

\[ 24b^4 - 12b^9 \]

Final Answer