Question

section 3.5
score: 1/6 answered: 1/6
question 2
find all zeros of the function f(x) = 8x³ - 26x² - 29x + 20. enter the zeros separated by commas.
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Explanation:

Step1: Try Rational Root Theorem

Possible rational roots are factors of 20 over factors of 8: $\pm1, \pm2, \pm4, \pm5, \pm10, \pm20, \pm\frac{1}{2}, \pm\frac{5}{2}, \pm\frac{1}{4}, \pm\frac{5}{4}, \pm\frac{1}{8}, \pm\frac{5}{8}$. Test $x = 4$: $f(4)=8(64)-26(16)-29(4)+20=512 - 416 - 116 + 20=0$. So $x - 4$ is a factor.

Step2: Perform Polynomial Division

Divide $8x^3 - 26x^2 - 29x + 20$ by $x - 4$. Using synthetic division or long division, we get $8x^2 + 6x - 5$.

Step3: Solve Quadratic Equation

Solve $8x^2 + 6x - 5 = 0$ using quadratic formula $x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}$, where $a = 8$, $b = 6$, $c=-5$. Discriminant: $36 + 160 = 196$. So $x=\frac{-6\pm14}{16}$. This gives $x=\frac{-6 + 14}{16}=\frac{8}{16}=\frac{1}{2}$ and $x=\frac{-6 - 14}{16}=\frac{-20}{16}=-\frac{5}{4}$.

Answer:

$4,\frac{1}{2},-\frac{5}{4}$