Questions: Factor out the GCF of the three terms, then complete the factorization of 3x^3+9x^2-30x

Transcript text: Factor out the GCF of the three terms, then complete the factorization of $3 x^{3}+9 x^{2}-30 x$

Solution

Solution Steps

To factor the trinomial $3x^3 + 9x^2 - 30x$, we first identify the greatest common factor (GCF) of all the terms. Once the GCF is factored out, we factor the remaining quadratic expression.

Step 1: Identify the Expression

We start with the expression $3x^3 + 9x^2 - 30x$.

Step 2: Factor Out the GCF

The greatest common factor (GCF) of the terms $3x^3$, $9x^2$, and $-30x$ is $3x$. We factor this out: \[ 3x^3 + 9x^2 - 30x = 3x(x^2 + 3x - 10) \]

Step 3: Factor the Quadratic Expression

Next, we need to factor the quadratic expression $x^2 + 3x - 10$. We look for two numbers that multiply to $-10$ and add to $3$. These numbers are $5$ and $-2$. Thus, we can factor the quadratic as: \[ x^2 + 3x - 10 = (x - 2)(x + 5) \]

Step 4: Combine the Factors

Now, we can combine the factors: \[ 3x(x^2 + 3x - 10) = 3x(x - 2)(x + 5) \]