1. **State the problem:** We want to analyze the sign of the function $$f(x) = x^3 + 2x^2 - 13x + 10$$ by creating a sign chart. 2. **Find the roots of the polynomial:** To create a sign chart, first find the roots where $$f(x) = 0$$. 3. **Try rational root theorem candidates:** Possible rational roots are factors of 10 over factors of 1: $$\pm1, \pm2, \pm5, \pm10$$. 4. **Evaluate at candidates:** - $$f(1) = 1 + 2 - 13 + 10 = 0$$, so $$x=1$$ is a root. - Divide $$f(x)$$ by $$x-1$$: $$f(x) = (x-1)(x^2 + 3x - 10)$$ 5. **Factor quadratic:** $$x^2 + 3x - 10 = (x+5)(x-2)$$ 6. **Roots are:** $$x = 1, -5, 2$$. 7. **Sign chart intervals:** - $$(-\infty, -5)$$ - $$(-5, 1)$$ - $$(1, 2)$$ - $$(2, \infty)$$ 8. **Test signs in each interval:** - For $$x < -5$$, pick $$x = -6$$: $$f(-6) = (-6-1)(-6+5)(-6-2) = (-7)(-1)(-8) = -56 < 0$$ - For $$-5 < x < 1$$, pick $$x=0$$: $$f(0) = (0-1)(0+5)(0-2) = (-1)(5)(-2) = 10 > 0$$ - For $$1 < x < 2$$, pick $$x=1.5$$: $$f(1.5) = (1.5-1)(1.5+5)(1.5-2) = (0.5)(6.5)(-0.5) = -1.625 < 0$$ - For $$x > 2$$, pick $$x=3$$: $$f(3) = (3-1)(3+5)(3-2) = (2)(8)(1) = 16 > 0$$ 9. **Summary of sign chart:** $$\begin{array}{c|cccccc} x & -\infty & & -5 & & 1 & & 2 & & +\infty \\ f(x) & - & 0 & + & 0 & - & 0 & + \\\end{array}$$ 10. **Interpretation:** The function is negative on $$(-\infty, -5)$$ and $$(1, 2)$$, positive on $$(-5, 1)$$ and $$(2, \infty)$$, and zero at $$x = -5, 1, 2$$. **Final answer:** The sign chart for $$f(x) = x^3 + 2x^2 - 13x + 10$$ is as above with roots at $$-5, 1, 2$$ and alternating signs accordingly.