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. (if possible, use the graphing zeros.)
$f(x) = $

Explanation:

Step1: Factor by grouping

Group the terms: \((4x^3 - 3x^2)+(100x - 75)\)
Factor out common terms: \(x^2(4x - 3)+25(4x - 3)\)
Now factor out \((4x - 3)\): \((4x - 3)(x^2 + 25)\)

Step2: Find zeros from factors

Set each factor equal to zero:

• For \(4x - 3 = 0\), solve for \(x\): \(4x = 3\) so \(x=\frac{3}{4}\)
• For \(x^2 + 25 = 0\), solve for \(x\): \(x^2=-25\) so \(x = \pm 5i\)

Step3: Write as product of linear factors

Using the zeros, the linear factors are \((x - \frac{3}{4})\), \((x - 5i)\), and \((x + 5i)\). But we can also write the first factor with integer coefficients by multiplying the first factor by 4 (and adjusting the other factors to keep the product the same). So \(f(x)=(4x - 3)(x - 5i)(x + 5i)\)

Answer:

(Zeros):
\(\frac{3}{4}, 5i, -5i\)