Question

find all the zeros of the function. (enter your answers as a comma - separated list)
$f(x)=4x^{3}-3x^{2}+100x - 75$
$x = $
write the polynomial as a product of linear factors. use a graphing utility to verify your results graphically.
$f(x)=$

Explanation:

Step1: Try rational roots

Use Rational Root Theorem: possible roots are $\pm\frac{1}{1},\pm\frac{3}{1},\pm\frac{5}{1},\pm\frac{15}{1},\pm\frac{25}{1},\pm\frac{75}{1},\pm\frac{1}{2},\pm\frac{3}{2},\pm\frac{5}{2},\pm\frac{15}{2},\pm\frac{25}{2},\pm\frac{75}{2},\pm\frac{1}{4},\pm\frac{3}{4},\pm\frac{5}{4},\pm\frac{15}{4},\pm\frac{25}{4},\pm\frac{75}{4}$. Test $x = \frac{3}{4}$: $f(\frac{3}{4}) = 4(\frac{3}{4})^3 - 3(\frac{3}{4})^2 + 100(\frac{3}{4}) - 75 = 4(\frac{27}{64}) - 3(\frac{9}{16}) + 75 - 75 = \frac{27}{16} - \frac{27}{16} + 0 = 0$. So $x = \frac{3}{4}$ is a root.

Step2: Factor the polynomial

Divide $f(x)$ by $(x - \frac{3}{4})$ or use synthetic division. Alternatively, factor by grouping: $f(x) = 4x^3 - 3x^2 + 100x - 75 = x^2(4x - 3) + 25(4x - 3) = (4x - 3)(x^2 + 25)$.

Step3: Find remaining roots

Solve $x^2 + 25 = 0$: $x^2 = -25 \implies x = \pm 5i$.

Answer:

$\frac{3}{4}, 5i, -5i$