In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=2x³+x²−3x+1

Verified step by step guidance

1. Identify the polynomial function: $f(x)=2x^3+x^2-3x+1$.
2. List all possible rational zeros using the Rational Root Theorem. The possible rational zeros are of the form $\backslash pm\; \backslash frac\{p\}\{q\}$, where $p$ divides the constant term (1) and $q$ divides the leading coefficient (2). So, possible rational zeros are $\backslash pm\; 1,\; \backslash pm\; \backslash frac\{1\}\{2\}$.
3. Use synthetic division to test each possible rational zero. Start by testing $x=1$: set up synthetic division with coefficients $2,\; 1,\; -3,\; 1$ and divide by $x-1$. Check the remainder to see if it is zero, indicating a root.
4. Once you find a zero (say $r$), use the quotient polynomial from the synthetic division (which will be a quadratic) to find the remaining zeros.
5. Solve the quadratic quotient either by factoring, completing the square, or using the quadratic formula to find the remaining zeros of the polynomial.

Key Concepts

Rational Root Theorem

The Rational Root Theorem helps identify all possible rational zeros of a polynomial by considering factors of the constant term and the leading coefficient. These possible roots are expressed as ±(factors of constant term)/(factors of leading coefficient), providing a finite list to test.

Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a binomial of the form (x - c). It simplifies calculations to test whether a candidate root is an actual zero by checking if the remainder is zero, and it produces a quotient polynomial for further analysis.

Factoring Polynomials and Finding Zeros

Once a zero is found using synthetic division, the quotient polynomial can be factored further or solved using other methods to find remaining zeros. This step breaks down the polynomial into simpler factors, revealing all roots including real and complex zeros.