QuestionRewrite $f(x)$ in factored form, but do not simplify: $f(x)=\frac{3 x^{3}+10 x^{2}-25 x}{3 x^{2}+17 x+20}=$

Studdy Solution

STEP 1

1. We are given a rational function $f(x) = \frac{3x^3 + 10x^2 - 25x}{3x^2 + 17x + 20}$ .
2. The goal is to express both the numerator and the denominator in factored form.
3. We will not simplify the expression after factoring.

STEP 2

1. Factor the numerator $3x^3 + 10x^2 - 25x$ .
2. Factor the denominator $3x^2 + 17x + 20$ .

STEP 3

Factor the numerator $3x^3 + 10x^2 - 25x$ .
First, factor out the greatest common factor (GCF) from the numerator:
$3x^3 + 10x^2 - 25x = x(3x^2 + 10x - 25)$
Next, factor the quadratic $3x^2 + 10x - 25$ . This can be done using the quadratic formula or by trial and error for integer roots. Assume it factors as:
$3x^2 + 10x - 25 = (3x - 5)(x + 5)$
Thus, the numerator in factored form is:
$x(3x - 5)(x + 5)$

STEP 4

Factor the denominator $3x^2 + 17x + 20$ .
Assume it factors as:
$3x^2 + 17x + 20 = (3x + 5)(x + 4)$

STEP 5

Rewrite $f(x)$ in factored form using the factored numerator and denominator:
$f(x) = \frac{x(3x - 5)(x + 5)}{(3x + 5)(x + 4)}$

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