A. verify the given factor(s) of the function f, b. find the...

a. verify the given factor$(s)$ of the function $f$, b. find the remaining factors of $f$, c. use your results to write the complete factorization of $f,$ and d. list all real zeros of $f$. Confirm your results by using a graphing utility to graph the function. $$\begin{array}{cc}\text{Function} && \text{Factor(s)} \\ f(x)=x^{4}-4 x^{3}-15 x^{2}+58 x-40 && (x-5),(x+4) \end{array}$$

a. verify the given factor$(s)$ of the function $f$,
b. find the remaining factors of $f$,
c. use your results to write the complete factorization of $f,$ and
d. list all real zeros of $f$. Confirm your results by using a graphing utility to graph the function.
$$\begin{array}{cc}\text{Function} && \text{Factor(s)} \\ f(x)=x^{4}-4 x^{3}-15 x^{2}+58 x-40 && (x-5),(x+4) \end{array}$$

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Key Concepts

Real Zeros of a Polynomial

The real zeros of a polynomial are the values of x for which the polynomial equals zero. Identifying these zeros involves solving the factored equations and observing their multiplicity, and they provide critical information about the graph of the polynomial.

Complete Factorization

Complete factorization involves expressing a polynomial as a product of its irreducible factors, often combining various methods such as verifying factors, division, and further factorization of lower degree polynomials.

Graphing Utility Verification

Using a graphing utility to verify the factorization involves plotting the polynomial function to visually confirm that the identified zeros are correct and that the graph intersects the x-axis at these points. This serves as an important check for the correctness of the algebraic factorization.

Factor Theorem

The Factor Theorem states that if a polynomial f(x) has f(a)=0, then (x?a) is a factor of f(x). This concept is used to verify whether a given binomial factor divides the polynomial without a remainder.

Polynomial Division

Polynomial division, including synthetic division or long division, is used to divide a polynomial by a known factor. This process simplifies the polynomial by reducing its degree and helps in finding the remaining factors.

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