Recall the formula to factor a difference of squares.
a^2- b^2 ⇔ ( a+ b)( a- b)
We can apply this formula to our expression.
( 5x)^2- 1^2 ⇔ ( 5x+ 1)( 5x- 1)
Checking Our Answer
Check your answer âś“
We can apply the Distributive Property and compare the result with the given expression.
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!
b To factor the given expression, we will first identify and factor out the greatest common factor. Then, we will use the formula for the difference of squares.
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. The GCF of the given expression is 5x.
Recall the formula to factor a difference of squares.
a^2- b^2 ⇔ ( a+ b)( a- b)
We can apply this formula to our expression.
5x ( x^2- 5^2 ) ⇔ 5x( x+ 5)( x- 5)
Checking Our Answer
Check your answer âś“
We can apply the Distributive Property and compare the result with the given expression.
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!
c 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+x-72 ⇒ x^2+x+- 72
In this case, we have -72. 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 -72 or -1 and 72
-72
2 and -36 or -2 and 36
-72
3 and -24 or -3 and 24
-72
4 and -18 or -4 and 18
-72
6 and -12 or -6 and 12
-72
8 and -9 or -8 and 9
-72
Next, let's consider the coefficient of the linear term.
x^2+1x- 72
For this term, we need the sum of the factors that produced the constant term to equal the coefficient of the linear term, 1. Looking at our table we can see that it is possible only for pairs from the last row. Let's check to see which of them is the correct pair.
Factors
Sum of Factors
8 and -9
-1
-8 and 9
1
We found the factors whose product is -72 and whose sum is 1.
x^2+1x- 72 ⇔ (x-8)(x+9)
Checking Our Answer
Check your answer âś“
We can check our answer by applying the Distributive Property and comparing the result with the given expression.
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!
d 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 x.
The result of factoring out a GCF from the given expression is a trinomial with a leading coefficient of one.
x( x^2-3x-18)
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^2-3x- 18
In this case, we have -18. 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 -18 or -1 and 18
-18
2 and -9 or -2 and 9
-18
3 and -6 or -3 and 6
-18
Next, let's consider the coefficient of the linear term.
x^2- 3x- 18
For this term, we need the sum of the factors that produced the constant term to equal the coefficient of the linear term, -3.
Factors
Sum of Factors
1 and -18
-17
-1 and 18
17
2 and -9
-7
-2 and 9
7
3 and -6
-3
We found the factors whose product is -18 and whose sum is -3.
x^2- 3x- 18 ⇔ (x+3)(x-6)
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.
x(x+3)(x-6)
Checking Our Answer
Check your answer âś“
We can check our answer by applying the Distributive Property and comparing the result with the given expression.
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!
