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.
### 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.
