Question

factor the polynomial $f(x)=x^2 - 2x - 24$.
$f(x)=\square$
the zeros of the parabola are:
$x=\square$ or $x=\square$
the vertex of the parabola is:
enter coordinates with parentheses, e.g., (x,y).
\square
graph the zeros and the vertex, then graph the parabola.

Explanation:

Step1: Factor the quadratic polynomial

Find two numbers that multiply to $-24$ and add to $-2$. These numbers are $-6$ and $4$.
$f(x)=(x-6)(x+4)$

Step2: Find the zeros

Set each factor equal to 0 and solve for $x$.
For $x-6=0$: $x=6$
For $x+4=0$: $x=-4$

Step3: Find vertex x-coordinate

Use vertex x-formula $x=-\frac{b}{2a}$ for $ax^2+bx+c$.
$x=-\frac{-2}{2(1)}=1$

Step4: Find vertex y-coordinate

Substitute $x=1$ into $f(x)$.
$f(1)=(1)^2-2(1)-24=1-2-24=-25$

Answer:

$f(x)=(x-6)(x+4)$
The zeros of the parabola are:
$x=6$ or $x=-4$
The vertex of the parabola is:
$(1, -25)$

For the graph:

1. Plot the zeros at the points $(6, 0)$ and $(-4, 0)$.
2. Plot the vertex at the point $(1, -25)$.
3. Draw a U-shaped parabola passing through these three points, opening upwards (since the coefficient of $x^2$ is positive).