Math Snap

STEP 1

What is this asking?
We need to break down this expression $15x^2 - 27x - 6$ into smaller pieces multiplied together, kind of like finding the prime factors of a number, but with xs!
Watch out!
Don't forget to look for common factors first!
It makes everything easier.
Also, double-check your answer by multiplying the factors back together.
It should match the original expression.

STEP 2

1. Find the Greatest Common Factor (GCF)
2. Factor the Trinomial

STEP 3

Let's look at our expression: $15x^2 - 27x - 6$.
We want to find the biggest number that divides evenly into all three coefficients (15, -27, and -6).
Think about what numbers go into all of them.

STEP 4

Looks like 3 is the greatest common factor!
So, we can factor out a 3 from each term: $15x^2 - 27x - 6 = 3(5x^2 - 9x - 2)$.
See how we divided each term by 3?
Now, we're left with a simpler trinomial inside the parentheses.

STEP 5

Now, we need to factor $5x^2 - 9x - 2$.
We're looking for two binomials that multiply together to give us this trinomial.
Since the x squared term has a coefficient of 5, we know one binomial will start with $5x$ and the other with $x$.
So, we have $(5x \ \ \ )(x \ \ \ )$.

STEP 6

Next, let's look at the constant term, which is -2.
We need two numbers that multiply to -2.
Our options are 1 and -2, or -1 and 2.

STEP 7

Now comes the tricky part!
We need to place these numbers in our binomials so that when we multiply everything out, the x term becomes -9$x$.
Let's try 1 and -2. $(5x + 1)(x - 2)$ gives us $5x^2 - 10x + x - 2 = 5x^2 - 9x - 2$.
BOOM! We found it!

So, the completely factored form of $15x^2 - 27x - 6$ is $3(5x + 1)(x - 2)$.
Remember, we factored out the 3 at the beginning!
Don't leave it out of the final answer.