Question
Which cubic function has zeros at x=0
with a multiplicity of 1, x=2
with multiplicity of 2, and x=-1
with a multiplicity of 1?
A
f(x)=4x(x+1)(x−2)2f(x)=4x(x+1)(x-2)^2f(x)=4x(x+1)(x−2)
2
B
g(x)=−2x(x−2)(x+1)g(x)=−2x(x-2)(x+1)g(x)=−2x(x−2)(x+1)
C
h(x)=−(x−1)(x+2)2h(x)=−(x-1)(x+2)^2h(x)=−(x−1)(x+2)
2
D
k(x)=2(x+1)2(x−2)2k(x)=2(x+1)^2(x-2)^2k(x)=2(x+1)
2
(x−2)
2
Answers
GPT-4o mini
To construct the cubic function with the given roots and their multiplicities, we can use the information about the zeros provided:
- A zero at \(x = 0\) with multiplicity 1 means the factor is \(x^1 = x\).
- A zero at \(x = 2\) with multiplicity 2 means the factor is \((x - 2)^2\).
- A zero at \(x = -1\) with multiplicity 1 means the factor is \((x + 1)^1 = (x + 1)\).
Thus, the function can be written as:
\[
f(x) = k \cdot x \cdot (x - 2)^2 \cdot (x + 1)
\]
where \(k\) is a constant.
Combining these factors, the function becomes:
\[
f(x) = k \cdot x \cdot (x - 2)^2 \cdot (x + 1)
\]
Let's analyze the provided options:
- **Option A:** \(f(x) = 4x(x + 1)(x - 2)^2\): This function includes the correct factors but includes an extra factor of \((x + 1)\) that we don't need. This option is incorrect.
- **Option B:** \(g(x) = -2x(x - 2)(x + 1)\): This option does not include the correct multiplicity for \((x - 2)\) (it should be squared). This option is incorrect.
- **Option C:** \(h(x) = -(x - 1)(x + 2)^2\): This option does not have the correct roots at all (it suggests roots of 1 and -2), so that is incorrect.
- **Option D:** \(k(x) = 2(x + 1)^2(x - 2)^2\): This option also does not match the required multiplicities, as it includes squared multiplicities for both \((x + 1)\) and \((x - 2)\).
None of the given options correspond to the cubic function with the specified zeros and multiplicities.
If we were to create the function, the correct form would look something like:
\[
f(x) = k \cdot x \cdot (x - 2)^2 \cdot (x + 1)
\]
where \(k\) can be any non-zero constant. If you have to choose one, a function of the form \(x \cdot (x - 2)^2 \cdot (x + 1)\) expanded would be:
\[
f(x) = x(x^2 - 4x + 4)(x + 1)
\]
We see that none of the provided functions meet the required criteria.
- A zero at \(x = 0\) with multiplicity 1 means the factor is \(x^1 = x\).
- A zero at \(x = 2\) with multiplicity 2 means the factor is \((x - 2)^2\).
- A zero at \(x = -1\) with multiplicity 1 means the factor is \((x + 1)^1 = (x + 1)\).
Thus, the function can be written as:
\[
f(x) = k \cdot x \cdot (x - 2)^2 \cdot (x + 1)
\]
where \(k\) is a constant.
Combining these factors, the function becomes:
\[
f(x) = k \cdot x \cdot (x - 2)^2 \cdot (x + 1)
\]
Let's analyze the provided options:
- **Option A:** \(f(x) = 4x(x + 1)(x - 2)^2\): This function includes the correct factors but includes an extra factor of \((x + 1)\) that we don't need. This option is incorrect.
- **Option B:** \(g(x) = -2x(x - 2)(x + 1)\): This option does not include the correct multiplicity for \((x - 2)\) (it should be squared). This option is incorrect.
- **Option C:** \(h(x) = -(x - 1)(x + 2)^2\): This option does not have the correct roots at all (it suggests roots of 1 and -2), so that is incorrect.
- **Option D:** \(k(x) = 2(x + 1)^2(x - 2)^2\): This option also does not match the required multiplicities, as it includes squared multiplicities for both \((x + 1)\) and \((x - 2)\).
None of the given options correspond to the cubic function with the specified zeros and multiplicities.
If we were to create the function, the correct form would look something like:
\[
f(x) = k \cdot x \cdot (x - 2)^2 \cdot (x + 1)
\]
where \(k\) can be any non-zero constant. If you have to choose one, a function of the form \(x \cdot (x - 2)^2 \cdot (x + 1)\) expanded would be:
\[
f(x) = x(x^2 - 4x + 4)(x + 1)
\]
We see that none of the provided functions meet the required criteria.