Lesson 1: Greatest Common Factor and Factor by Grouping

New Concept

This lesson introduces factoring, the reverse of multiplication. You'll master finding the Greatest Common Factor (GCF) and use it to factor polynomials and apply the factor by grouping method, essential tools for simplifying expressions.

What’s next

Next, you'll master finding the GCF through interactive examples and apply this skill in practice cards on factoring polynomials and grouping.

Property

The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.

HOW TO: Find the Greatest Common Factor (GCF) of two expressions.
Step 1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
Step 2. List all factors—matching common factors in a column. In each column, circle the common factors.
Step 3. Bring down the common factors that all expressions share.
Step 4. Multiply the factors.

Examples

• Find the GCF of $42$ and $70$. We factor each number: $42 = 2 \cdot 3 \cdot 7$ and $70 = 2 \cdot 5 \cdot 7$. The common factors are $2$ and $7$. So, the GCF is $2 \cdot 7 = 14$.
• Find the GCF of $15a^2$ and $25a^3$. We factor each term: $15a^2 = 3 \cdot 5 \cdot a \cdot a$ and $25a^3 = 5 \cdot 5 \cdot a \cdot a \cdot a$. The common factors are $5, a, a$. The GCF is $5a^2$.
• Find the GCF of $12x^2y$ and $18xy^2$. We factor each term: $12x^2y = 2 \cdot 2 \cdot 3 \cdot x \cdot x \cdot y$ and $18xy^2 = 2 \cdot 3 \cdot 3 \cdot x \cdot y \cdot y$. The common factors are $2, 3, x, y$. The GCF is $6xy$.

Property

Distributive Property
If $a, b, c$ are real numbers, then

The form on the left is used to multiply. The form on the right is used to factor.

HOW TO: Factor the greatest common factor from a polynomial.
Step 1. Find the GCF of all the terms of the polynomial.
Step 2. Rewrite each term as a product using the GCF.
Step 3. Use the "reverse" Distributive Property to factor the expression.
Step 4. Check by multiplying the factors.

Examples

• Factor $7y + 21$. The GCF of $7y$ and $21$ is $7$. We rewrite the expression as $7 \cdot y + 7 \cdot 3$. Using the reverse distributive property, we get $7(y+3)$.
• Factor $6a^3 - 18a^2 + 12a$. The GCF of all terms is $6a$. We rewrite this as $6a \cdot a^2 - 6a \cdot 3a + 6a \cdot 2$. Factoring out the GCF gives $6a(a^2 - 3a + 2)$.
• Factor $-5x - 20$. When the leading term is negative, we factor out a negative GCF. The GCF is $-5$. This gives $-5(x) + (-5)(4)$, which factors to $-5(x+4)$.

Property

When there is no common factor of all the terms of a polynomial, look for a common factor in just some of the terms. When there are four terms, a good way to start is by separating the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find a common factor emerges from both parts.

HOW TO: Factor by grouping.
Step 1. Group terms with common factors.
Step 2. Factor out the common factor in each group.
Step 3. Factor the common factor from the expression.
Step 4. Check by multiplying the factors.

Examples

• Factor $ab + 5a + 3b + 15$. Group the terms: $(ab + 5a) + (3b + 15)$. Factor the GCF from each group: $a(b+5) + 3(b+5)$. Factor out the common binomial: $(b+5)(a+3)$.
• Factor $x^2 + 2x - 5x - 10$. Group the terms: $(x^2 + 2x) + (-5x - 10)$. Factor GCFs: $x(x+2) - 5(x+2)$. Factor out the common binomial: $(x+2)(x-5)$.
• Factor $mn - 8m + 4n - 32$. Group the terms: $(mn - 8m) + (4n - 32)$. Factor GCFs: $m(n-8) + 4(n-8)$. Factor out the common binomial: $(n-8)(m+4)$.