Lesson 38: Simplifying Expressions Using the GCF

New Concept

Finding the GCF means finding the largest monomial that divides without a remainder into each term of a polynomial.

What’s next

Next, you'll apply this concept to factor polynomials and simplify algebraic fractions, transforming them into their most basic forms.

Property

Writing a composite number as a product of only prime numbers. For example, the prime factorization of 6 is $2 \cdot 3$.

Examples

$120 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5$
$924 = 2 \cdot 2 \cdot 3 \cdot 7 \cdot 11$

Explanation

Think of it as cracking a code to find a number's prime building blocks! You can use a factor tree or division by primes to find these core ingredients. No matter how you start, you'll always end with the same unique set of prime factors for any given number.

Property

The greatest common factor (GCF) is the product of the greatest integer and the greatest power of each variable that divides without a remainder into each term.

Examples

For $6a^2b^3 + 8a^4b^2c$, the GCF is $2a^2b^2$.
For $8c^3d^2e$ and $12c^2d^4e^2$, the GCF is $4c^2d^2e$.

Explanation

Basically, the GCF is the biggest, beefiest monomial that fits perfectly into every term. To find it, break each term down to its prime factors, identify all the common parts, and multiply them together.

Property

Factoring a polynomial is the inverse of the Distributive Property. It rewrites a sum or difference of monomials as a product of factors.

Examples

$6x^3 + 8x^2 - 2x = 2x(3x^2 + 4x - 1)$
$9x^4y^2 - 9x^5y = 9x^4y(y - x^2)$

Explanation

Think of this as 'un-distributing'! You find the GCF of all the terms, pull it out to the front, and write what’s left over inside parentheses. It's a great way to simplify complex expressions.