Question

for each function below:
a) write the function in factored form
b) find the zeros of the function

1. $f(x) = x^3 - 3x^2 - 8x + 24$

Explanation:

Part (a): Write the function in Factored Form

Step 1: Group terms

Group the first two terms and the last two terms of the polynomial:
$f(x)=(x^{3}-3x^{2})+(-8x + 24)$

Step 2: Factor out GCF from each group

Factor out $x^{2}$ from the first group and $-8$ from the second group:
$f(x)=x^{2}(x - 3)-8(x - 3)$

Step 3: Factor out common binomial factor

Notice that both terms have a common factor of $(x - 3)$, so we factor that out:
$f(x)=(x - 3)(x^{2}-8)$

To find the zeros, we set $f(x) = 0$ and solve for $x$:

Step 1: Set factored form equal to zero

Using the factored form $f(x)=(x - 3)(x^{2}-8)=0$ (or using the more factored form with square roots)

Step 2: Solve each factor for zero

• For the first factor: $x - 3=0\implies x = 3$
• For the second factor $x^{2}-8=0$:

Add $8$ to both sides: $x^{2}=8$
Take square roots: $x=\pm\sqrt{8}=\pm2\sqrt{2}$

Answer:

(Factored Form):
$f(x)=(x - 3)(x^{2}-8)$ or we can further factor $x^{2}-8$ as a difference of squares (if we consider real numbers) to get $f(x)=(x - 3)(x-\sqrt{8})(x + \sqrt{8})=(x - 3)(x - 2\sqrt{2})(x + 2\sqrt{2})$

Part (b): Find the zeros of the function