Let's start by finding the GCF of 3 and 3. Since these are the same numbers, we can identify the GCF as 3. To find the GCF of the variables, we need to identify the variables repeated in both terms and write them with their minimum exponents.
\begin{aligned}
\textbf{Factors of }\bm{1^\text{st}}\textbf{ Variable:}&\ x, {\color{#FF0000}{x^2}}, x^3\\
\textbf{Factors of }\bm{2^\text{nd}}\textbf{ Variable:}&\ x, {\color{#FF0000}{x^2}}
\end{aligned}
We see that there is one repeated variable factor, x^2.
Thus, the GCF of the expression is 3* x^2= 3x^2. Now we can write the given expression in terms of the GCF.
3x^3-3x^2 ⇔ 3x^2* x- 3x^2* 1
Finally, we will factor out the GCF.
3x^2* x- 3x^2* 1 ⇔ 3x^2(x+1)
b Let's start factoring by first identifying the 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 2.
The result of factoring out a GCF from the given expression is a trinomial with a leading coefficient of 1.
2( x^2-5x+6)
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^2-5x+6
In this case we have 6. 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 6
6
-1 and -6
6
2 and 3
6
-2 and -3
6
Next, let's consider the coefficient of the linear term.
x^2-5x+6
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 6
7
-1 and -6
-7
2 and 3
5
-2 and -3
-5
We found the factors whose product is 6 and whose sum is -5.
x^2-5x+6 ⇔ (x-2)(x-3)
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-2)(x-3)
c We want to factor the given equation. Let's start factoring by first identifying the GCF.
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 8.
The result of factoring out a GCF from the given expression is a binomial with a leading coefficient of 1.
8( x^2-4)
Let's temporarily only focus on this binomial, and we will bring back the GCF after factoring.
Factor the Expression
Do you notice that the binomial is a difference of two perfect squares? This can be factored using the difference of squares method.
a^2 - b^2 ⇔ (a+b)(a-b)
To do so, we first need to express each term as a perfect square.
Expression
x^2-4
Rewrite as Perfect Squares
x^2 - 2^2
Apply the Formula
(x+2)(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.
8(x+2)(x-2)
d We want to completely factor the given expression. To do so, we will first identify and factor out the GCF.
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. The GCF of the given expression is 2x.
Here we have a quadratic trinomial of the form ax^2+bx+c, where |a| ≠1 and there are no common factors. To factor this expression, we will rewrite the middle term, bx, as two terms. The coefficients of these two terms will be factors of ac whose sum must be b.
2x( 2x^2+5x-12 )
⇕
2x( 2x^2+5x+(- 12) )
We have that a= 2, b=5, and c=- 12. There are now three steps we need to follow in order to rewrite the above expression.
Find a c. Since we have that a= 2 and c=- 12, the value of a c is 2* (- 12)=- 24.
Find factors of a c. Since ac=- 24, which is negative, we need factors of a c to have opposite signs — one positive and one negative. Since b=5, which is positive, the absolute value of the positive factor will need to be greater than the absolute value of the negative factor so that their sum is positive.
Rewrite bx as two terms. Now that we know which factors are the ones to be used, we can rewrite bx as two terms.
2x(2x^2+5x-12 )
⇕
2x ( 2x^2 - 3x + 8x-12 )
Finally, we will factor the last expression obtained.
