Alright, let's go through the detailed, step-by-step process for constructing a function that meets the given criteria:
1. Degree of the Function: We know the function must have a degree of either 2, 3, 4, or 5. For this example, let's work with a function of degree 3.
2. Zero with Multiplicity of 2: The function must have at least one zero (root) that has a multiplicity of 2. This means that if [tex]\( r \)[/tex] is this zero, then the factor [tex]\((x - r)\)[/tex] should appear squared in the polynomial.
3. Choosing the Roots: Let's suppose the roots of the polynomial are:
- [tex]\( r_1 = 2 \)[/tex] with multiplicity 2
- [tex]\( r_2 = -1 \)[/tex] with multiplicity 1
4. Constructing the Polynomial:
- Given the roots and their multiplicities, the polynomial can be written in factored form as:
[tex]\[
f(x) = (x - 2)^2 \cdot (x + 1)
\][/tex]
5. Expanding the Polynomial:
- To convert this polynomial to its expanded form, we perform the multiplication:
[tex]\[
f(x) = (x - 2)^2 \cdot (x + 1)
\][/tex]
- First, expand [tex]\((x - 2)^2\)[/tex]:
[tex]\[
(x - 2)^2 = x^2 - 4x + 4
\][/tex]
- Now, multiply this by [tex]\((x + 1)\)[/tex]:
[tex]\[
(x^2 - 4x + 4)(x + 1) = x^3 + x^2 - 4x^2 - 4x + 4x + 4 = x^3 - 3x^2 + 4
\][/tex]
6. Conclusion:
- The polynomial in its factored form is:
[tex]\[
f(x) = (x - 2)^2 \cdot (x + 1)
\][/tex]
- The polynomial in its expanded form is:
[tex]\[
f(x) = x^3 - 3x^2 + 4
\][/tex]
- The roots of the polynomial are:
- 2 (with multiplicity 2)
- -1 (with multiplicity 1)
- The degree of the polynomial is 3.
To summarize, the polynomial function that meets the given criteria can be expressed in different forms, and the key properties of this function are as follows:
- Factored Form: [tex]\( f(x) = (x - 2)^2 \cdot (x + 1) \)[/tex]
- Expanded Form: [tex]\( f(x) = x^3 - 3x^2 + 4 \)[/tex]
- Roots of the Polynomial: [tex]\( 2, 2, -1 \)[/tex]
- Multiplicities of the Roots: [tex]\( 2, 1, 1 \)[/tex]
- Degree of the Polynomial: [tex]\( 3 \)[/tex]
