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Answer :
To determine which given expressions are equivalent to [tex]\(\left(4 x^2+\frac{1}{2} x^3\right) \cdot (2 x+6-\frac{5}{2} x^2)\)[/tex], we'll multiply the expressions step-by-step and simplify the result.
First, expand the product:
[tex]\[
\left(4 x^2+\frac{1}{2} x^3\right) \cdot (2 x+6-\frac{5}{2} x^2)
\][/tex]
Distribute each term in [tex]\(\left(4 x^2+\frac{1}{2} x^3\right)\)[/tex] across each term in [tex]\( (2 x+6-\frac{5}{2} x^2) \)[/tex]:
[tex]\[
= (4 x^2 \cdot 2 x) + (4 x^2 \cdot 6) + (4 x^2 \cdot -\frac{5}{2} x^2) + \left(\frac{1}{2} x^3 \cdot 2 x\right) + \left(\frac{1}{2} x^3 \cdot 6\right) + \left(\frac{1}{2} x^3 \cdot -\frac{5}{2} x^2\right)
\][/tex]
Simplify each term:
[tex]\[
= 8 x^3 + 24 x^2 - 10 x^4 + x^4 + 3 x^3 + 8 x^3 + 24 x^2
\][/tex]
Combine like terms:
[tex]\[
= -\frac{5}{4} x^5 - 9 x^4 + 11 x^3 + 24 x^2
\][/tex]
Now, let's compare this result with each provided option:
1. [tex]\( -\frac{5}{4} x^5 - 9 x^4 + 11 x^3 + 24 x^2 \)[/tex]
- Matches exactly.
2. [tex]\( -\frac{5}{4} x^5 + 5 x^4 + 5 x^3 + 60 x^2 \)[/tex]
- Does not match.
3. [tex]\( -\frac{5}{4} x^5 - 10 x^4 + x^4 + 3 x^3 + 8 x^3 + 24 x^2 \)[/tex]
- Matches after combining terms: [tex]\( -\frac{5}{4} x^5 - 9 x^4 + 11 x^3 + 24 x^2 \)[/tex].
4. [tex]\( -\frac{5}{4} x^5 - 9 x^4 + 8 x^3 + 3 x^3 + 24 x^2 \)[/tex]
- Matches after combining terms: [tex]\( -\frac{5}{4} x^5 - 9 x^4 + 11 x^3 + 24 x^2 \)[/tex].
5. [tex]\( -\frac{5}{4} x^5 + x^4 - 10 x^4 + 11 x^3 + 24 x^2 \)[/tex]
- Matches after combining terms: [tex]\( -\frac{5}{4} x^5 - 9 x^4 + 11 x^3 + 24 x^2 \)[/tex].
Thus, the following expressions are equivalent to [tex]\(\left(4 x^2+\frac{1}{2} x^3\right) \cdot (2 x+6-\frac{5}{2} x^2)\)[/tex]:
[tex]\[
-\frac{5}{4} x^5-9 x^4+11 x^3+24 x^2
\][/tex]
[tex]\[
-\frac{5}{4} x^5-10 x^4+x^4+3 x^3+8 x^3+24 x^2
\][/tex]
[tex]\[
-\frac{5}{4} x^5-9 x^4+8 x^3+3 x^3+24 x^2
\][/tex]
[tex]\[
-\frac{5}{4} x^5+x^4-10 x^4+11 x^3+24 x^2
\][/tex]
First, expand the product:
[tex]\[
\left(4 x^2+\frac{1}{2} x^3\right) \cdot (2 x+6-\frac{5}{2} x^2)
\][/tex]
Distribute each term in [tex]\(\left(4 x^2+\frac{1}{2} x^3\right)\)[/tex] across each term in [tex]\( (2 x+6-\frac{5}{2} x^2) \)[/tex]:
[tex]\[
= (4 x^2 \cdot 2 x) + (4 x^2 \cdot 6) + (4 x^2 \cdot -\frac{5}{2} x^2) + \left(\frac{1}{2} x^3 \cdot 2 x\right) + \left(\frac{1}{2} x^3 \cdot 6\right) + \left(\frac{1}{2} x^3 \cdot -\frac{5}{2} x^2\right)
\][/tex]
Simplify each term:
[tex]\[
= 8 x^3 + 24 x^2 - 10 x^4 + x^4 + 3 x^3 + 8 x^3 + 24 x^2
\][/tex]
Combine like terms:
[tex]\[
= -\frac{5}{4} x^5 - 9 x^4 + 11 x^3 + 24 x^2
\][/tex]
Now, let's compare this result with each provided option:
1. [tex]\( -\frac{5}{4} x^5 - 9 x^4 + 11 x^3 + 24 x^2 \)[/tex]
- Matches exactly.
2. [tex]\( -\frac{5}{4} x^5 + 5 x^4 + 5 x^3 + 60 x^2 \)[/tex]
- Does not match.
3. [tex]\( -\frac{5}{4} x^5 - 10 x^4 + x^4 + 3 x^3 + 8 x^3 + 24 x^2 \)[/tex]
- Matches after combining terms: [tex]\( -\frac{5}{4} x^5 - 9 x^4 + 11 x^3 + 24 x^2 \)[/tex].
4. [tex]\( -\frac{5}{4} x^5 - 9 x^4 + 8 x^3 + 3 x^3 + 24 x^2 \)[/tex]
- Matches after combining terms: [tex]\( -\frac{5}{4} x^5 - 9 x^4 + 11 x^3 + 24 x^2 \)[/tex].
5. [tex]\( -\frac{5}{4} x^5 + x^4 - 10 x^4 + 11 x^3 + 24 x^2 \)[/tex]
- Matches after combining terms: [tex]\( -\frac{5}{4} x^5 - 9 x^4 + 11 x^3 + 24 x^2 \)[/tex].
Thus, the following expressions are equivalent to [tex]\(\left(4 x^2+\frac{1}{2} x^3\right) \cdot (2 x+6-\frac{5}{2} x^2)\)[/tex]:
[tex]\[
-\frac{5}{4} x^5-9 x^4+11 x^3+24 x^2
\][/tex]
[tex]\[
-\frac{5}{4} x^5-10 x^4+x^4+3 x^3+8 x^3+24 x^2
\][/tex]
[tex]\[
-\frac{5}{4} x^5-9 x^4+8 x^3+3 x^3+24 x^2
\][/tex]
[tex]\[
-\frac{5}{4} x^5+x^4-10 x^4+11 x^3+24 x^2
\][/tex]
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