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)=2x^3+x^2−3x+1

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1. Identify the polynomial function: \( f(x) = 2x^3 + x^2 - 3x + 1 \).
2. List all possible rational zeros using the Rational Root Theorem, which states that any rational zero, \( \frac{p}{q} \), is a factor of the constant term (1) divided by a factor of the leading coefficient (2). Thus, the possible rational zeros are \( \pm 1, \pm \frac{1}{2} \).
3. Use synthetic division to test each possible rational zero. Start with \( x = 1 \) and perform synthetic division to see if it results in a remainder of 0.
4. If \( x = 1 \) is a zero, the quotient from synthetic division will be a quadratic polynomial. Use this quotient to find the remaining zeros by factoring or using the quadratic formula.
5. Continue testing other possible rational zeros if necessary, and use the quotient to find all zeros of the polynomial function.

Key Concepts

Rational Root Theorem

The Rational Root Theorem provides a method for identifying all possible rational zeros of a polynomial function. It states that any rational solution, expressed as a fraction p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient. For the polynomial f(x)=2x^3+x^2−3x+1, this theorem helps in listing potential rational zeros to test.

Synthetic Division

Synthetic division is a simplified form of polynomial long division used to divide a polynomial by a linear factor of the form (x - c). It allows for efficient computation of the quotient and remainder, making it easier to test potential rational zeros. If the remainder is zero, then (x - c) is a factor of the polynomial, confirming c as a zero of the function.

Finding Remaining Zeros

Once an actual zero is found using synthetic division, the polynomial can be expressed as a product of the linear factor corresponding to that zero and a reduced polynomial. The remaining zeros can then be found by factoring or using the quadratic formula on the reduced polynomial. This process is essential for fully solving the polynomial equation and identifying all its roots.