Write a polynomial f(x) that meets the given conditions....

Write a polynomial $f(x)$ that meets the given conditions. Answers may vary. (See Example 10 )
Degree 4 polynomial with zeros $1, \frac{3}{2}$ (each with multiplicity 1 ), and 0 (with multiplicity 2).

Key Concepts

Factorization

Factorization involves expressing a polynomial as a product of its linear (or irreducible) factors, each corresponding to a root and its multiplicity. This technique is essential for solving polynomial equations, understanding their structure, and simplifying expressions.

Multiplicity

Multiplicity refers to the number of times a particular zero appears in a polynomial. A zero with a multiplicity greater than one indicates that the corresponding factor appears that many times in the factored form, affecting the graph's touch or cross behavior at that zero.

Zeros or Roots

The zeros or roots of a polynomial are the values of the variable that make the polynomial equal to zero. These values are critical in understanding the behavior of a function and are used to factorize polynomials, which aids in solving equations and analyzing graphs.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in its expression when the polynomial is expressed in its standard form. This concept is important as it influences the shape of the graph, the number of roots, and properties like the end-behavior of the function.

Polynomial Function

A polynomial function is an expression constructed from variables and coefficients using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. These functions form the cornerstone of algebra and provide a way to model many kinds of mathematical relationships and phenomena.

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