Question

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)=2x3−3x2−11x+6

Accepted Answer

Identify the polynomial function: $$f(x) = 2x$$^{3}$$ - 3x$$^{2}$$ - 11x + 6. $$List all possible rational zeros using the Rational Root Theorem. These are all fractions $$\frac{p}{q}$$ where $$p$$ divides the constant term (6) and $$q$$ divides the leading coefficient (2). So, possible values of $$p$$ are $$\pm1, \pm2, \pm3, \pm6$$ and possible values of $$q$$ are $$\pm1, \pm2. $$Therefore, the possible rational zeros are $$\pm1, \pm2, \pm3, \pm6, \pm\frac{1}{2}, \pm\frac{3}{2}. $$Use synthetic division to test each possible rational zero by substituting them into the polynomial. Start with one candidate zero, perform synthetic division, and check if the remainder is zero. If the remainder is zero, that candidate is an actual zero of the polynomial. Once an actual zero is found, use the quotient polynomial from the synthetic division (which will be of degree 2) to find the remaining zeros. This can be done by factoring the quadratic or using the quadratic formula if necessary. Summarize the zeros found: the actual zero from synthetic division and the zeros from the quadratic factor, which together give all zeros of the original cubic polynomial.

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)=2x³−3x²−11x+6

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Key Concepts

Rational Root Theorem

Synthetic Division

Factoring Polynomials and Finding Zeros