Question

f(x) = 3x³ + 5x² - 11x + 3; q. find all zeros. (hint: graph)

Explanation:

Step1: Test rational root candidates

Using Rational Root Theorem, possible roots: $\pm1, \pm3, \pm\frac{1}{3}$
Test $x=1$: $f(1)=3(1)^3+5(1)^2-11(1)+3=3+5-11+3=0$. So $(x-1)$ is a factor.

Step2: Perform polynomial division

Divide $3x^3+5x^2-11x+3$ by $(x-1)$:
$$\frac{3x^3+5x^2-11x+3}{x-1}=3x^2+8x-3$$

Step3: Factor quadratic

Factor $3x^2+8x-3$:
$3x^2+8x-3=(3x-1)(x+3)$

Step4: Solve for zeros

Set each factor to 0:
$x-1=0 \implies x=1$
$3x-1=0 \implies x=\frac{1}{3}$
$x+3=0 \implies x=-3$

Answer:

$x=1$, $x=-3$, $x=\frac{1}{3}$