Factorial Expressions
Factoring an expression means writing a numerical or algebraic expression as a product of factors. To factor expressions, we can make use of the distributive property.
How can we Factorize an expression?
Step 1: Find the prime factors of the given expression.
Step 2: Encircle the common factors and find the GCF.
Step 3: Write each term of the expression as a product of the GCF
and the remaining factor.
Step 4: Use the distributive property and simplify the expression.
Let us recall the distributive property.
Solved Examples
Example 1: Factorize the numerical expressions.
(a) 18 + 12
(b) 30 – 20
Solution:
Part (a)
First, we have to find the factors of 18 and 12.
The factors of 18 are 1, 2, 3, 6, 9, 18
The factors of 12 are 1, 2, 3, 4, 6, 12
The greatest common factor (GCF) is 6.
Now, as a product of the GCF and the remaining factor, write each term of the expression. Then factor the expression using the distributive property.
18 + 12 = 6(3) + 6(2)
= 6 (3 + 2) Using the distributive property
Part (b)
First, we have to find the factors of 30 and 20.
The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30
The factors of 20 are 1, 2, 4, 5, 10, 20
The greatest common factor (GCF) is 5.
Now, as a product of the GCF and the remaining factor, write each term of the expression. Then factor the expression using the distributive property.
30 – 20 = 5(6) – 5(4)
= 5(6 – 4) Using the distributive property
Example 2: Solve the algebra equation by simplifying the expression and using the GCF property.
(a) 3a + 30
(b) 25x – 10y
Solution:
Part (a)
We have: 3a + 30
First, we have to write the prime factorization of a given number to find the GCF.
The prime factorization of 3a is 3\(\times\)a
The prime factorization of 30 is 2\(\times\)3\(\times\)5
The greatest common factor ( GCF ) is 3.
Now, we have to form an expression using the GCF of the given expression.
3a + 30 = 3(a) + 3(10)
= 3(a + 10) Using the distributive property
Part (b)
We have: 25x – 10y
First, we have to write the prime factorization of a given number to find the GCF.
The prime factorization of 25x is 5\(\times\)5\(\times\)x
The prime factorization of 10y is 2\(\times\)5\(\times\)y
The greatest common factor (GCF) is 5.
Now, we have to form an expression using the GCF of the given expression.
25x-10y = 5(5x) + 5(2y)
= 5(5x + 2y) Using the distributive property
Example 3: Each book you purchase for your electronic reader, qualifies for a discount. Each book was originally priced at x dollars. For a total of (5x – 15) dollars, you purchase five books. Multiply the expression by two. What conclusions can you draw about the discount?
Solution:
We have: 5x – 15
First, we have to write the prime factorization of a given expression to find the GCF.
The prime factorization of 5x is 5\(\times\)x
The prime factorization of 15 is 3\(\times\)5
The greatest common factor (GCF) is 5.
Now, we have to form an expression using the GCF of the given expression.
5x – 15 = 5(x) – 5(3)
= 5(x – 3) Using the distributive property
The number of books purchased is represented by the factor 5. The discounted price of each book is represented by the factor (x – 3).
This factor is the difference between the two terms, indicating that each book’s original price, $x, has been reduced by $3.
As a result, the factored expression indicates a $3 discount for each book that is purchased. A total saving of $15 is shown in the original expression.
A factorial function is a mathematical formula represented by an exclamation mark “!”. The factorial is the product of all integers less than or equal to n but greater than or equal to 1 for an integer n greater than or equal to 1. The factorial value of 0 is 1 by definition.
Simply use the formula to find the factorial of 5, or 5!, by multiplying all the integers together from 5 to 1. We get 120 when we use the formula to find 5!. As a result, 5! = 120.
The product of all integers from 1 to that number is the factorial of that number. The factorial of 6 is 1*2*3*4*5*6 = 720, for example. Negative numbers have no factorial, and the factorial of zero is one, 0! = 1.