Question

In Exercises 49–50, find all the zeros of each polynomial function and write the polynomial as a product of linear factors. f(x)=2x4+3x3+3x−2f(x) = $$2x^4$$ + $$3x^3 + 3x - 2$$

Accepted Answer

First, write down the polynomial function: $$f(x) = 2x^4$ + 3x^3$ + 3x - 2. $$Look for possible rational zeros using the Rational Root Theorem. The possible rational zeros are of the form $$\pm \frac{p}{q}$$, where $$p$$ divides the constant term $$-2$$ and $$q$$ divides the leading coefficient $$2. So $$possible zeros are $$\pm 1$$, $$\pm 2$$, $$\pm \frac{1}{2}. $$Test each possible rational zero by substituting into $$f(x) or by $$using synthetic division to check if it yields a remainder of zero. When you find a zero, say $$r$$, factor out $$(x - r)$$ from the polynomial. After factoring out one linear factor, divide the original polynomial by this factor to get a cubic polynomial. Repeat the process of finding zeros and factoring until the polynomial is expressed as a product of linear factors. Once all zeros are found and the polynomial is factored completely, write the polynomial as $$f(x) = a(x - r_1$)(x - r_2$)(x - r_3$)(x - r_4$)$$, where $$a is $$the leading coefficient and $$r_1$$, $$r_2$$, $$r_3$$, $$r_4$$ are the zeros.

In Exercises 49–50, find all the zeros of each polynomial function and write the polynomial as a product of linear factors. $f (x) = 2 x^{4} + 3 x^{3} + 3 x - 2$

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Key Concepts

Polynomial Zeros

Factoring Polynomials

Rational Root Theorem and Synthetic Division

In Exercises 49–50, find all the zeros of each polynomial function and write the polynomial as a product of linear factors. f(x)=2x4+3x3+3x−2f(x) = 2x^4 + 3x^3 + 3x - 2