Question

a) choose the correct roots of the polynomial equation: $x^3 + 4x^2 - 11x - 30 = (x + 2)(x - 3)(x + 5) = 0$ $\bigcirc$ $x = 4, -11, -30$ $\bigcirc$ $x = 1, -2, -15$ $\bigcirc$ $x = -2, 3, -5$ $\bigcirc$ $x = 2, -3, -5$ b) where does the graph of the polynomial function, $f(x) = x^2 + 5x + 6$, cross the $x$-axis? $\bigcirc$ $x = 6, 0$ $\bigcirc$ $x = -5, -6$ $\bigcirc$ $x = -2, -3$ $\bigcirc$ $x = 2, 3$ 7) find the roots of the polynomial equation $(2x + 8)(8x - 5) = 0$, $\square$ $\frac{5}{8}$ $\square$ $3$ $\square$ $-5$ $\square$ $-\frac{3}{2}$ $\square$ $2$

Explanation:

Step1: Solve for roots using zero product property

For $(x+2)(x-3)(x+5)=0$, set each factor to 0:
$x+2=0 \implies x=-2$; $x-3=0 \implies x=3$; $x+5=0 \implies x=-5$

Step2: Factor quadratic to find x-intercepts

For $f(x)=x^2+5x+6$, factor as $(x+2)(x+3)=0$. Set factors to 0:
$x+2=0 \implies x=-2$; $x+3=0 \implies x=-3$

Step3: Solve linear factors for roots

For $(2x+3)(3x-5)=0$, set each factor to 0:
$2x+3=0 \implies x=-\frac{3}{2}$; $3x-5=0 \implies x=\frac{5}{3}$

Answer:

1. $x=-2, 3, -5$
2. $x=-2, -3$
3. $\frac{5}{3}$, $-\frac{3}{2}$