• Learn to simplify expressions using the greatest common factor or GCF.

• Simplifying expression that contain numbers often requires knowledge of prime number and factors.
• Recall that a prime number is a whole number that is only divisible by itself and 1.
• All whole numbers other than 1 that are not prime are composite numbers.
• Composite numbers have whole-number factors other than 1 and the number itself.
• They can be written as a product of prime numbers, which is called prime factorization .

• Several methods can be used to find the prime factorization of a number.
• The process requires breaking down the composite umbers until all the factors are prime.
• The Prime factorization for number 24 can be found in at least three ways.

Example
The Prime factorization for number 24 can be found in at least three ways.
• It does not matter which method is used to find a prime factorization.

Finding the Prime Factorization of a Number
Find the prime factorization of the number.
120

Finding the Prime Factorization of a Number
Find the prime factorization of the number.
924

Find the prime factorization of the number.
100

Find the prime factorization of the number.
51

• Prime factorization can be used when determining the Greatest Common Factor (GFC) of Monomial .
• The GFC of a monomial is the product of the greatest integer that divides without remainder into the coefficients and the greatest power of each variable that divides without a remainder into each term.
• Finding the GCF means finding the larges monomial that divides without a remainder into each term of a polynomial.

Determining the GCF of Algebraic Expession
Find the GCF of the expression.
6a 2 b 3 + 8a 2 b 2 c

Determining the GCF of Algebraic Expression
Find the GCF of the expression.
24m 3 n 4 + 32mn 5 p

Find the GCF of the expression.
8c 4 d 2 e – 12c 3 d 4 e 2

Find the GCF of the expression.
5p 2 q 5 r 2 – 10 pq 2 r 2

• Finding the GCF of a polynomial allows you to factor it and to write the polynomial as a product of factors instead of the sum or difference of monomials.
• Factoring a polynomial is the inverse of the
Distributive Property.
• Using the Distributive Property will “undo” the factoring of the GCF.

Factoring a Polynomial
Factor the polynomial completely.
6x 3 + 8x 2 – 2x

Factor the polynomial completely.
8d 2 e 3 + 12d 3 e 2

Factoring a Polynomial
Factor the polynomial completely.
9x 4 y 2 – 9x 6 y

Factor the polynomial completely.
12x 4 y 2 z – 42x 3 y 3 z 2

• Fractions can be simplified if the numerator and denominator contain common factors.
• This is because the operations of multiplication and division undo each other.
• An algebraic fraction can only be simplified if the numerator and the denominator have common factors.

Simplifying Algebraic Fractions
Simplify the expression
3p + 3
3

Simplify the expression
6x + 18
6

Simplifying Algebraic Fractions
Simplify the expression
5x – 25x 2
5xy

Simplify the expression
18x + 45x 3
9x

Application: Finding the Height of an Object
The formula h = –16t 2 + 72t + 12 can be used to represent the height of an object that is launched into the air from 12 feet off the ground with an initial velocity of 72 feet/second.
Rewrite the formula by factoring the right side using the GCF and making the t 2 –term positive.

The formula h = –16t 2 + 60t + 4 can be used to find the height of an object that is launched into the air from 4 feet off the ground with an initial velocity of 60 feet/second. Rewrite the formula by factoring the right side using the GCF and making the t 2 –term positive.