(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. $f(x)=2 x^{3}+x^{2}-5 x+2 \quad(x+2),(x-1)$

Key Concepts

Remainder Theorem

The Remainder Theorem is closely related to the Factor Theorem and states that when a polynomial f(x) is divided by (x - c), the remainder is f(c). This allows for a quick check to see if (x - c) is a factor (i.e., if the remainder is zero) and aids in verifying potential factors.

Zeros of a Polynomial

Zeros (or roots) of a polynomial are the solutions to the equation f(x) = 0. Identifying these zeros is crucial because they correspond to the x-intercepts of the polynomial’s graph. Knowing the zeros not only helps in the factorization of the polynomial but also in understanding the behavior of the function.

Factor Theorem

The Factor Theorem states that a polynomial f(x) has a factor (x - c) if and only if f(c) = 0. This theorem is fundamental in polynomial factorization because it provides a direct method for testing whether a given linear expression is a factor of the polynomial by simply evaluating the polynomial at the corresponding value.

Polynomial Division

Polynomial division, including long division and synthetic division, is a method used to divide polynomials. When one factor is known, dividing the polynomial by that factor reduces the polynomial’s degree, which helps in finding the remaining factors. This technique is essential for simplifying problems and finding complete factorizations.

Graphical Verification

Graphical verification involves using graphing utilities or software to plot the polynomial function and visually confirm the locations of its zeros. This process serves as an additional check on the algebraic factorization and helps verify that the computed zeros align with the points where the graph intersects the x-axis.

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