The 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 2.
The result of factoring out a GCF from the given expression is a trinomial with a leading coefficient of 1.
2( x^2-x-2)
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 1, think of the process as multiplying two binomials in reverse. Let's start by taking a look at the constant term.
x^2-x- 2
In this case, we have -2. This is a negative number, so for the product of the constant terms in the factors to be negative, these constants must have the opposite sign (one positive and one negative.)
Factor Constants
Product of Constants
1 and - 2
- 2
-1 and 2
- 2
Next, let's consider the coefficient of the linear term.
x^2-1x- 2
For this term, we need the sum of the factors that produced the constant term to equal the coefficient of the linear term, -1.
Factors
Sum of Factors
1 and - 2
-1
-1 and 2
1
We found the factors whose product is - 2 and whose sum is -1.
x^2-1x-2 ⇔ (x+1)(x-2)
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.
2(x+1)(x-2)
b Let's start factoring by first identifying the greatest common factor (GCF). Then, we will rewrite the expression as a trinomial with a leading coefficient of 1.
Factor Out The GCF
The 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 4.
The result of factoring out a GCF from the given expression is a trinomial with a leading coefficient of 1.
4( x^2-6x+9)
Let's temporarily only focus on this trinomial, and we will bring back the GCF after factoring.
Factor the Expression
We want to factor the given expression. Factoring is much easier when our polynomial is a perfect square trinomial. To determine if an expression is a perfect square trinomial, we need to ask ourselves three questions.
Is the first term a perfect square?
x^2= x^2 ✓
Is the last term a perfect square?
9= 3^2 ✓
Is the middle term twice the product of 3 and x?
6x=2* 3* x ✓
As we can see, the answer to all three questions is yes! Therefore, we can write the trinomial as the square of a binomial. Note there is an subtraction sign in the middle.
( x^2-6x+9) ⇔ ( x- 3)^2
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.
4(x-3)^2
