1. **State the problem:** We are given the polynomial function $$f(x) = x^3 - 5x^2 - 4x + 20$$ and need to: a. List all possible rational zeros using the Rational Zero Theorem. b. Use synthetic division to find one actual rational zero. c. Use that zero to find all zeros of the polynomial. 2. **List possible rational zeros (Rational Zero Theorem):** The Rational Zero Theorem states that any rational zero, expressed as $$\frac{p}{q}$$, must have $$p$$ as a factor of the constant term and $$q$$ as a factor of the leading coefficient. - Constant term: 20 - Leading coefficient: 1 Factors of 20: $$\pm1, \pm2, \pm4, \pm5, \pm10, \pm20$$ Factors of 1: $$\pm1$$ Possible rational zeros are all $$\pm$$ factors of 20 divided by factors of 1, so: $$\pm1, \pm2, \pm4, \pm5, \pm10, \pm20$$ 3. **Use synthetic division to test possible zeros:** Try $$x=1$$: Coefficients: 1 (for $$x^3$$), -5 (for $$x^2$$), -4 (for $$x$$), 20 (constant) Synthetic division setup: 1 | 1 -5 -4 20 | 1 -4 -8 ---------------- 1 -4 -8 12 (remainder) Remainder is 12, so $$x=1$$ is not a zero. Try $$x=2$$: 2 | 1 -5 -4 20 | 2 -6 -20 ---------------- 1 -3 -10 0 (remainder) Remainder is 0, so $$x=2$$ is a zero. 4. **Use zero $$x=2$$ to factor the polynomial:** The quotient from synthetic division is: $$x^2 - 3x - 10$$ Factor this quadratic: $$x^2 - 3x - 10 = (x - 5)(x + 2)$$ 5. **Find all zeros:** Zeros are: $$x = 2, 5, -2$$ **Final answers:** - a. Possible rational zeros: $$\pm1, \pm2, \pm4, \pm5, \pm10, \pm20$$ - b. One rational zero is $$2$$ - c. All zeros are $$2, 5, -2$$