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Instructions: Multiply the following monomials by polynomials. Type your answers in standard form.

Problem A:
[tex]\[
\begin{array}{l}
-3x^3\left(5x^2 - 3x + 7\right) \\
(-3x^3)(5x^2) + (-3x^3)(-3x) + (-3x^3)(7) \\
= -15x^5 + 9x^4 - 21x^3
\end{array}
\][/tex]

Problem B:
[tex]\[
\begin{array}{l}
3x\left(x^2 - 4x + 5\right) \\
(3x)(x^2) + (3x)(-4x) + (3x)(5) \\
= 3x^3 - 12x^2 + 15x
\end{array}
\][/tex]



Answer :

GinnyAnswer
Let's work through Problem B step by step, using the same methods as we did for Problem A.

### Problem B
Given:
[tex]\[ 2a^2b(b^3 - 4ab + 5a^2) \][/tex]

We need to multiply each term inside the parenthesis by the monomial [tex]\(2a^2b\)[/tex].

Here's the process in detail:

1. Distribute [tex]\(2a^2b\)[/tex] to each term in the polynomial [tex]\(b^3 - 4ab + 5a^2\)[/tex]:

[tex]\[ 2a^2b(b^3 - 4ab + 5a^2) = (2a^2b \cdot b^3) + (2a^2b \cdot -4ab) + (2a^2b \cdot 5a^2) \][/tex]

2. Multiply [tex]\(2a^2b\)[/tex] by each term individually:

- First term: [tex]\(2a^2b \cdot b^3\)[/tex]
[tex]\[ 2a^2b \cdot b^3 = 2a^2b^{1+3} = 2a^2b^4 \][/tex]

- Second term: [tex]\(2a^2b \cdot -4ab\)[/tex]
[tex]\[ 2a^2b \cdot -4ab = 2 \cdot -4 \cdot a^2 \cdot a \cdot b \cdot b = -8a^{2+1}b^{1+1} = -8a^3b^2 \][/tex]

- Third term: [tex]\(2a^2b \cdot 5a^2\)[/tex]
[tex]\[ 2a^2b \cdot 5a^2 = 2 \cdot 5 \cdot a^2 \cdot a^2 \cdot b = 10a^{2+2}b = 10a^4b \][/tex]

3. Write the final expression in standard form:

[tex]\[ 2a^2b(b^3 - 4ab + 5a^2) = 2a^2b^4 - 8a^3b^2 + 10a^4b \][/tex]

So the answer for Problem B is:
[tex]\[ 2a^2b^4 - 8a^3b^2 + 10a^4b \][/tex]

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