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

Verified step by step guidance

1. First, write down the polynomial function: \(f(x) = 2x^4 + 3x^3 + 3x - 2\).
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}$.
3. 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.
4. 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.
5. 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.

Key Concepts

Polynomial Zeros

Zeros of a polynomial are the values of x for which the polynomial equals zero. Finding these roots helps in understanding the behavior of the function and is essential for factoring the polynomial into linear factors.

Factoring Polynomials

Factoring involves expressing a polynomial as a product of simpler polynomials, ideally linear factors. This process often uses the zeros of the polynomial, allowing it to be written as (x - r) for each root r.

Rational Root Theorem and Synthetic Division

The Rational Root Theorem helps identify possible rational zeros by considering factors of the constant and leading coefficients. Synthetic division is a streamlined method to test these candidates and simplify the polynomial for further factoring.