1. **State the problem:** Find all zeros of the polynomial function $$f(x) = 2x^4 - 9x^3 - 12x^2 + 29x + 30$$. 2. **Recall the goal:** We want to find all values of $x$ such that $$f(x) = 0$$. 3. **Use the Rational Root Theorem:** Possible rational roots are factors of the constant term 30 divided by factors of the leading coefficient 2. Possible roots: $$\pm1, \pm2, \pm3, \pm5, \pm6, \pm10, \pm15, \pm30, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{5}{2}, \pm\frac{15}{2}$$. 4. **Test possible roots by substitution or synthetic division:** - Test $x=2$: $$f(2) = 2(2)^4 - 9(2)^3 - 12(2)^2 + 29(2) + 30 = 2(16) - 9(8) - 12(4) + 58 + 30 = 32 - 72 - 48 + 58 + 30 = 0$$ So, $x=2$ is a root. 5. **Divide $f(x)$ by $(x-2)$ using synthetic division:** Coefficients: 2, -9, -12, 29, 30 Bring down 2. Multiply 2*2=4, add to -9 = -5. Multiply -5*2 = -10, add to -12 = -22. Multiply -22*2 = -44, add to 29 = -15. Multiply -15*2 = -30, add to 30 = 0. Quotient polynomial: $$2x^3 - 5x^2 - 22x - 15$$ 6. **Factor the cubic $2x^3 - 5x^2 - 22x - 15$:** Try rational roots again: possible roots are factors of 15 over factors of 2. Test $x=3$: $$2(3)^3 - 5(3)^2 - 22(3) - 15 = 2(27) - 5(9) - 66 - 15 = 54 - 45 - 66 - 15 = -72 \neq 0$$ Test $x=-1$: $$2(-1)^3 - 5(-1)^2 - 22(-1) - 15 = -2 - 5 + 22 - 15 = 0$$ So, $x=-1$ is a root. 7. **Divide the cubic by $(x+1)$:** Coefficients: 2, -5, -22, -15 Bring down 2. Multiply 2*(-1) = -2, add to -5 = -7. Multiply -7*(-1) = 7, add to -22 = -15. Multiply -15*(-1) = 15, add to -15 = 0. Quotient polynomial: $$2x^2 - 7x - 15$$ 8. **Factor the quadratic $2x^2 - 7x - 15$:** Find two numbers that multiply to $2 \times (-15) = -30$ and add to $-7$. These are $-10$ and $3$. Rewrite: $$2x^2 - 10x + 3x - 15 = 0$$ Group: $$(2x^2 - 10x) + (3x - 15) = 0$$ Factor: $$2x(x - 5) + 3(x - 5) = 0$$ $$ (2x + 3)(x - 5) = 0$$ 9. **Solve each factor:** $$2x + 3 = 0 \Rightarrow x = -\frac{3}{2}$$ $$x - 5 = 0 \Rightarrow x = 5$$ 10. **List all roots from smallest to largest:** $$x = -\frac{3}{2}, -1, 2, 5$$ No repeated roots found. **Final answer:** $$x = -\frac{3}{2}, -1, 2, 5$$