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The function $g$ is defined for all numbers $x$ by $g(x)=r x^{\wedg
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25 Jul 2025, 21:15
25 Jul 2025, 21:15
Expert Reply
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Question Stats:
80% (01:20) correct
20% (01:49) wrong based on 5 sessions
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The function $g$ is defined for all numbers $x$ by $\(g(x)=r x^5-s x^3+t\)$, where $r, s$, and $t$ are constants. If $\(g(1)=0\)$, which of the following could be the values of the three constants?
Indicate all such values.
\(\boxed{A}\) - $\(\mathrm{r}=3, \mathrm{~s}=2\)$ and $\(\mathrm{t}=-1\)$
\(\boxed{B}\) - $\(\mathrm{r}=-4, \mathrm{~s}=3\)$ and $\(\mathrm{t}=1\)$
\(\boxed{C}\) - $\(\mathrm{r}=-2, \mathrm{~s}=-1\)$ and $\(\mathrm{t}=1\)$
\(\boxed{D}\) - $\(\mathrm{r}=4, \mathrm{~s}=3\)$ and $\(\mathrm{t}=-7\)$
\(\boxed{E}\) - $\(\mathrm{r}=4, \mathrm{~s}=-3\)$ and $\(\mathrm{t}=-7\)$
Indicate all such values.
\(\boxed{A}\) - $\(\mathrm{r}=3, \mathrm{~s}=2\)$ and $\(\mathrm{t}=-1\)$
\(\boxed{B}\) - $\(\mathrm{r}=-4, \mathrm{~s}=3\)$ and $\(\mathrm{t}=1\)$
\(\boxed{C}\) - $\(\mathrm{r}=-2, \mathrm{~s}=-1\)$ and $\(\mathrm{t}=1\)$
\(\boxed{D}\) - $\(\mathrm{r}=4, \mathrm{~s}=3\)$ and $\(\mathrm{t}=-7\)$
\(\boxed{E}\) - $\(\mathrm{r}=4, \mathrm{~s}=-3\)$ and $\(\mathrm{t}=-7\)$
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Re: The function $g$ is defined for all numbers $x$ by $g(x)=r x^{\wedg
[#permalink]
03 Aug 2025, 22:27
03 Aug 2025, 22:27
Expert Reply
Given the function:
$$
\(g(x)=r x^5-s x^3+t\)
$$
where $r, s, t$ are constants.
We know:
$$
\(g(1)=0\)
$$
Substitute $x=1$ :
$$
\(g(1)=r(1)^5-s(1)^3+t=r-s+t=0\)
$$
This gives the equation:
$$
\(r-s+t=0\)
$$
Now, let's check each option to see which satisfy this equation.
Option A:
$$
\(r=3, \quad s=2, \quad t=-1\)
$$
Calculate:
$$
\(3-2+(-1)=3-2-1=0\)
$$
Satisfies the equation.
Option B:
$$
\(r=-4, \quad s=3, \quad t=1\)
$$
Calculate:
$$
\(-4-3+1=-4-3+1=-6 \neq 0\)
$$
Does not satisfy.
Option C:
$$
\(r=-2, \quad s=-1, \quad t=1\)
$$
Calculate:
$$
\(-2-(-1)+1=-2+1+1=0\)
$$
Satisfies.
Option D:
$$
\(r=4, \quad s=3, \quad t=-7\)
$$
Calculate:
$$
\(4-3-7=4-3-7=-6 \neq 0\)
$$
Does not satisfy.
Option E:
$$
\(r=4, \quad s=-3, \quad t=-7\)
$$
Calculate:
$$
\(4-(-3)-7=4+3-7=0\)
$$
Satisfies.
Final answer:
The options that satisfy $\(r-s+t=0\)$ are:
$A, C$, and $E$.
$$
\(g(x)=r x^5-s x^3+t\)
$$
where $r, s, t$ are constants.
We know:
$$
\(g(1)=0\)
$$
Substitute $x=1$ :
$$
\(g(1)=r(1)^5-s(1)^3+t=r-s+t=0\)
$$
This gives the equation:
$$
\(r-s+t=0\)
$$
Now, let's check each option to see which satisfy this equation.
Option A:
$$
\(r=3, \quad s=2, \quad t=-1\)
$$
Calculate:
$$
\(3-2+(-1)=3-2-1=0\)
$$
Satisfies the equation.
Option B:
$$
\(r=-4, \quad s=3, \quad t=1\)
$$
Calculate:
$$
\(-4-3+1=-4-3+1=-6 \neq 0\)
$$
Does not satisfy.
Option C:
$$
\(r=-2, \quad s=-1, \quad t=1\)
$$
Calculate:
$$
\(-2-(-1)+1=-2+1+1=0\)
$$
Satisfies.
Option D:
$$
\(r=4, \quad s=3, \quad t=-7\)
$$
Calculate:
$$
\(4-3-7=4-3-7=-6 \neq 0\)
$$
Does not satisfy.
Option E:
$$
\(r=4, \quad s=-3, \quad t=-7\)
$$
Calculate:
$$
\(4-(-3)-7=4+3-7=0\)
$$
Satisfies.
Final answer:
The options that satisfy $\(r-s+t=0\)$ are:
$A, C$, and $E$.
gmatclubot
Re: The function $g$ is defined for all numbers $x$ by $g(x)=r x^{\wedg [#permalink]
03 Aug 2025, 22:27
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