Answer :
Certainly! Let’s define functions that have the specified properties: varying degrees and at least one zero with a multiplicity of 2.
### Function 1: Degree 2
We want a quadratic function (degree 2) that has at least one zero with a multiplicity of 2.
- Function Notation: [tex]\( f(x) = (x - 2)^2 \)[/tex]
This function has a zero at [tex]\( x = 2 \)[/tex] with a multiplicity of 2. If you expand it:
[tex]\[ f(x) = (x - 2)(x - 2) = x^2 - 4x + 4 \][/tex]
### Function 2: Degree 3
We want a cubic function (degree 3) that has at least one zero with a multiplicity of 2.
- Function Notation: [tex]\( g(x) = (x + 1)^2(x - 3) \)[/tex]
This function has a zero at [tex]\( x = -1 \)[/tex] with a multiplicity of 2 and another zero at [tex]\( x = 3 \)[/tex]. If you expand it:
[tex]\[ g(x) = (x + 1)(x + 1)(x - 3) = (x^2 + 2x + 1)(x - 3) \][/tex]
### Function 3: Degree 4
We want a quartic function (degree 4) that has at least one zero with a multiplicity of 2.
- Function Notation: [tex]\( h(x) = (x - 4)^2(x + 2)(x - 1) \)[/tex]
This function has a zero at [tex]\( x = 4 \)[/tex] with a multiplicity of 2, another zero at [tex]\( x = -2 \)[/tex], and another zero at [tex]\( x = 1 \)[/tex]. If you expand it:
[tex]\[ h(x) = (x - 4)(x - 4)(x + 2)(x - 1) \][/tex]
### Function 4: Degree 5
We want a quintic function (degree 5) that has at least one zero with a multiplicity of 2.
- Function Notation: [tex]\( j(x) = (x - 5)^2(x + 3)(x - 2)(x + 1) \)[/tex]
This function has a zero at [tex]\( x = 5 \)[/tex] with a multiplicity of 2, and additional zeros at [tex]\( x = -3 \)[/tex], [tex]\( x = 2 \)[/tex], and [tex]\( x = 1 \)[/tex]. If you expand it:
[tex]\[ j(x) = (x - 5)(x - 5)(x + 3)(x - 2)(x + 1) \][/tex]
To summarize, the functions with the specific degrees and zeros with multiplicities of 2 are:
1. [tex]\( f(x) = (x - 2)^2 \)[/tex]
2. [tex]\( g(x) = (x + 1)^2(x - 3) \)[/tex]
3. [tex]\( h(x) = (x - 4)^2(x + 2)(x - 1) \)[/tex]
4. [tex]\( j(x) = (x - 5)^2(x + 3)(x - 2)(x + 1) \)[/tex]
These functions fulfill the conditions of having at least one zero with a multiplicity of 2 and ranging in degrees from 2 to 5.
### Function 1: Degree 2
We want a quadratic function (degree 2) that has at least one zero with a multiplicity of 2.
- Function Notation: [tex]\( f(x) = (x - 2)^2 \)[/tex]
This function has a zero at [tex]\( x = 2 \)[/tex] with a multiplicity of 2. If you expand it:
[tex]\[ f(x) = (x - 2)(x - 2) = x^2 - 4x + 4 \][/tex]
### Function 2: Degree 3
We want a cubic function (degree 3) that has at least one zero with a multiplicity of 2.
- Function Notation: [tex]\( g(x) = (x + 1)^2(x - 3) \)[/tex]
This function has a zero at [tex]\( x = -1 \)[/tex] with a multiplicity of 2 and another zero at [tex]\( x = 3 \)[/tex]. If you expand it:
[tex]\[ g(x) = (x + 1)(x + 1)(x - 3) = (x^2 + 2x + 1)(x - 3) \][/tex]
### Function 3: Degree 4
We want a quartic function (degree 4) that has at least one zero with a multiplicity of 2.
- Function Notation: [tex]\( h(x) = (x - 4)^2(x + 2)(x - 1) \)[/tex]
This function has a zero at [tex]\( x = 4 \)[/tex] with a multiplicity of 2, another zero at [tex]\( x = -2 \)[/tex], and another zero at [tex]\( x = 1 \)[/tex]. If you expand it:
[tex]\[ h(x) = (x - 4)(x - 4)(x + 2)(x - 1) \][/tex]
### Function 4: Degree 5
We want a quintic function (degree 5) that has at least one zero with a multiplicity of 2.
- Function Notation: [tex]\( j(x) = (x - 5)^2(x + 3)(x - 2)(x + 1) \)[/tex]
This function has a zero at [tex]\( x = 5 \)[/tex] with a multiplicity of 2, and additional zeros at [tex]\( x = -3 \)[/tex], [tex]\( x = 2 \)[/tex], and [tex]\( x = 1 \)[/tex]. If you expand it:
[tex]\[ j(x) = (x - 5)(x - 5)(x + 3)(x - 2)(x + 1) \][/tex]
To summarize, the functions with the specific degrees and zeros with multiplicities of 2 are:
1. [tex]\( f(x) = (x - 2)^2 \)[/tex]
2. [tex]\( g(x) = (x + 1)^2(x - 3) \)[/tex]
3. [tex]\( h(x) = (x - 4)^2(x + 2)(x - 1) \)[/tex]
4. [tex]\( j(x) = (x - 5)^2(x + 3)(x - 2)(x + 1) \)[/tex]
These functions fulfill the conditions of having at least one zero with a multiplicity of 2 and ranging in degrees from 2 to 5.
