Question

which expression is equivalent to x^3(2+y^5)

Answer 1

(1/2x^2-3)(4y^3+5x^2)

Answer 2

The expression \( x^3(2 + y^5) \) can be considered in its original form. However, if you're looking for an alternate representation by distributing \( x^3 \) over the terms inside the parentheses, you can expand it:

\[
x^3(2 + y^5) = x^3 \cdot 2 + x^3 \cdot y^5
\]

This can be written as:

\[
2x^3 + x^3y^5
\]

So, both \( x^3(2 + y^5) \) and \( 2x^3 + x^3y^5 \) are equivalent expressions.

Answer 3

To expand the expression \((\frac{1}{2}x^2 - 3)(4y^3 + 5x^2)\), we will apply the distributive property (also known as the FOIL method for binomials). We'll multiply each term in the first parentheses by each term in the second parentheses.

1. **Multiply \(\frac{1}{2}x^2\) by \(4y^3\)**:
\[
\frac{1}{2}x^2 \cdot 4y^3 = 2x^2y^3
\]

2. **Multiply \(\frac{1}{2}x^2\) by \(5x^2\)**:
\[
\frac{1}{2}x^2 \cdot 5x^2 = \frac{5}{2}x^4
\]

3. **Multiply \(-3\) by \(4y^3\)**:
\[
-3 \cdot 4y^3 = -12y^3
\]

4. **Multiply \(-3\) by \(5x^2\)**:
\[
-3 \cdot 5x^2 = -15x^2
\]

Now we can combine all these results to get the expanded form:

\[
2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2
\]

Thus, the fully expanded form of \((\frac{1}{2}x^2 - 3)(4y^3 + 5x^2)\) is:

\[
\frac{5}{2}x^4 + 2x^2y^3 - 15x^2 - 12y^3
\]