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