(a) verify the given factors of f(x) (b) find the remaining...

(a) verify the given factors of $f(x)$ (b) find the remaining factor(s) of $f(x)$ (c) use your results to write the complete factorization of $f(x),$ (d) list all real zeros of $f,$ and (e) confirm your results by using a graphing utility to graph the function. $$\begin{array}{ll}\text { Function } & \text { Factors } \\f(x)=3 x^{3}-x^{2}-8 x-4 & (x+1),(x-2)\end{array}$$

(a) verify the given factors of $f(x)$
(b) find the remaining factor(s) of $f(x)$
(c) use your results to write the complete factorization of $f(x),$
(d) list all real zeros of $f,$ and
(e) confirm your results by using a graphing utility to graph the function.
$$\begin{array}{ll}\text { Function } & \text { Factors } \\f(x)=3 x^{3}-x^{2}-8 x-4 & (x+1),(x-2)\end{array}$$

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Key Concepts

Graphing Utility for Verification

Graphing utilities provide a visual means to verify the results obtained from algebraic factorization. By plotting the polynomial function, one can confirm the location of the zeros and observe the general behavior of the function, ensuring consistency between analytical results and graphical representation.

Finding Zeros of a Polynomial

Once the polynomial is factored, each factor can be set equal to zero to find its roots or zeros. These zeros are the x-values where the function evaluates to zero, which is an essential step in completely factorizing the polynomial and understanding its behavior.

Factor Theorem

The Factor Theorem states that if a polynomial f(x) has a factor of the form (x - c), then f(c) must equal zero. This concept is used to verify whether a given value is a root of the polynomial, hence confirming that its corresponding linear factor is indeed a factor of the polynomial.

Polynomial Division

Polynomial division, including methods like long division or synthetic division, is a process to divide one polynomial by another. This is crucial for reducing the polynomial's degree once a factor is known, thereby assisting in finding the remaining factors of the polynomial.

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