Let's start factoring by first identifying the greatest common factor. Then, we will rewrite the expression as a trinomial with a leading coefficient of one.

Factor Out the GCF

The greatest common factor (GCF) of an expression is a common factor of the terms in the expression. It is the common factor with the greatest coefficient and the greatest exponent. In this case, the GCF is

3 .

3 x^{2} - 15 x + 12

SplitIntoFactors

Split into factors

3 \cdot x^{2} - 3 \cdot 5 n + 3 \cdot 4

FactorOut

Factor out $3$

3 (x^{2} - 5 x + 4)

The result of factoring out a GCF from the given expression is a trinomial with a leading coefficient of one.

3 (x^{2} - 5 x + 4)

Let's temporarily only focus on this trinomial, and we will bring back the GCF after factoring.

Factor the Expression

To factor a trinomial with a leading coefficient of one, think of the process as multiplying two binomials in reverse. Let's start by taking a look at the constant term.

x^{2} - 5 x + 4

In this case, we have

4 .

This is a positive number, so for the product of the constant terms in the factors to be positive, these constants must have the same sign (both positive or both negative.)

Factor Constants	Product of Constants
$1$ and $4$	$4$
$- 1$ and $- 4$	$4$
$2$ and $2$	$4$
$- 2$ and $- 2$	$4$

Next, let's consider the coefficient of the linear term.

x^{2} - 5 x + 4

For this term, we need the sum of the factors that produced the constant term to equal the coefficient of the linear term,

- 5 .

Factors	Sum of Factors
$1$ and $4$	$5$
$- 1$ and $- 4$	$- 5$
$2$ and $2$	$4$
$- 2$ and $- 2$	$- 4$

We found the factors whose product is

4

and whose sum is

- 5 .

x^{2} - 5 x + 4 \Leftrightarrow (x - 1) (x - 4)

Wait! Before we finish, remember that we factored out a GCF from the original expression. To fully complete the factored expression, let's reintroduce that GCF now.

3 (x - 1) (x - 4)

Checking Our Answer

Check your answer

✓

We can check our answer by applying the Distributive Property and comparing the result with the given expression.

3 (x - 1) (x - 4)

Distr

Distribute $3$

(3 x - 3) (x - 4)

Distr

Distribute $(x - 4)$

3 x (x - 4) - 3 (x - 4)

Distr

Distribute $3 x$

3 x^{2} - 12 x - 3 (n - 4)

Distr

Distribute $- 3$

3 x^{2} - 12 x - 3 x + 12

SubTerm

Subtract term

3 x^{2} - 15 x + 12

After applying the Distributive Property and simplifying, the result is the same as the given expression. Therefore, we can be sure our solution is correct!

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