College

We appreciate your visit to 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. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

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.

Thanks for taking the time to read 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. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada