College

Which are equivalent expressions for

[tex]\left(4 x^2+\frac{1}{2} x^3\right) \cdot\left(2 x+6-\frac{5}{2} x^2\right)[/tex]?

Select all that apply.

A. [tex]-\frac{5}{4} x^5-9 x^4+11 x^3+24 x^2[/tex]

B. [tex]-\frac{5}{4} x^5+5 x^4+5 x^3+60 x^2[/tex]

C. [tex]-\frac{5}{4} x^5-10 x^4+x^4+3 x^3+8 x^3+24 x^2[/tex]

D. [tex]-\frac{5}{4} x^5-9 x^4+8 x^3+3 x^3+24 x^2[/tex]

E. [tex]-\frac{5}{4} x^5+x^4-10 x^4+11 x^3+24 x^2[/tex]

Answer :

To determine which expressions are equivalent to multiplying [tex]\((4x^2 + \frac{1}{2}x^3) \cdot (2x + 6 - \frac{5}{2}x^2)\)[/tex], we'll verify each option by expanding the original expression and comparing it to the provided options.

### Steps to Solve:

1. Expand the Original Expression:

We want to multiply each term in the first expression with each term in the second expression:

[tex]\[
(4x^2 + \frac{1}{2}x^3) \cdot (2x + 6 - \frac{5}{2}x^2)
\][/tex]

2. Distribute the Terms:

- Distribute [tex]\(4x^2\)[/tex]:
- [tex]\(4x^2 \cdot 2x = 8x^3\)[/tex]
- [tex]\(4x^2 \cdot 6 = 24x^2\)[/tex]
- [tex]\(4x^2 \cdot -\frac{5}{2}x^2 = -10x^4\)[/tex]

- Distribute [tex]\(\frac{1}{2}x^3\)[/tex]:
- [tex]\(\frac{1}{2}x^3 \cdot 2x = x^4\)[/tex]
- [tex]\(\frac{1}{2}x^3 \cdot 6 = 3x^3\)[/tex]
- [tex]\(\frac{1}{2}x^3 \cdot -\frac{5}{2}x^2 = -\frac{5}{4}x^5\)[/tex]

3. Combine Like Terms:

[tex]\[
-\frac{5}{4}x^5 + (-10x^4 + x^4) + (8x^3 + 3x^3) + 24x^2
\][/tex]

Simplifying the expression, we get:

- Highest degree terms:
- [tex]\(-\frac{5}{4}x^5\)[/tex]
- [tex]\(x^4\)[/tex] terms:
- [tex]\(-10x^4 + x^4 = -9x^4\)[/tex]
- [tex]\(x^3\)[/tex] terms:
- [tex]\(8x^3 + 3x^3 = 11x^3\)[/tex]
- [tex]\(x^2\)[/tex] term:
- [tex]\(24x^2\)[/tex]

Result: [tex]\(-\frac{5}{4}x^5 - 9x^4 + 11x^3 + 24x^2\)[/tex]

4. Compare with Provided Options:

We need to check each of the given options to find which ones are equivalent:

- Option 1: [tex]\(-\frac{5}{4} x^5 - 9 x^4 + 11 x^3 + 24 x^2\)[/tex]
Equivalent, matches the expanded form.

- Option 2: [tex]\(-\frac{5}{4} x^5 + 5 x^4 + 5 x^3 + 60 x^2\)[/tex]
Not Equivalent, does not match.

- Option 3: [tex]\(-\frac{5}{4} x^5 - 10 x^4 + x^4 + 3 x^3 + 8 x^3 + 24 x^2\)[/tex]
Equivalent, correctly simplifies to the expanded form.

- Option 4: [tex]\(-\frac{5}{4} x^5 - 9 x^4 + 8 x^3 + 3 x^3 + 24 x^2\)[/tex]
Equivalent, correctly simplifies to the expanded form.

- Option 5: [tex]\(-\frac{5}{4} x^5 + x^4 - 10 x^4 + 11 x^3 + 24 x^2\)[/tex]
Equivalent, correctly simplifies to the expanded form.

So, the equivalent expressions are options 1, 3, 4, and 5.