Greatest Common Factor (GCF)

In algebra, simplifying expressions is a crucial skill that helps us solve complex problems more easily. One important technique in simplification is finding the largest common factor (LCF) of terms in an algebraic expression. This process allows us to factor out the greatest shared component, making the expression more compact and often easier to work with.

Factors are the building blocks of multiplication. They are the numbers that you can multiply together to produce another number. For example, [latex]2[/latex] and [latex]10[/latex] are factors of [latex]20[/latex], as are [latex]4, 5, 1, 20[/latex]. To factor a number is to rewrite it as a product. [latex]20=4\cdot{5}[/latex] or [latex]20=1\cdot{20}[/latex]. In algebra, we use the word factor as both a noun – something being multiplied – and as a verb – the action of rewriting a sum or difference as a product. Factoring is very helpful in simplifying expressions and solving equations involving polynomials.

Common factors are factors that are shared by two or more terms in an expression. The largest factor that is common to all terms in an expression is called the Greatest Common Factor (GCF).

• Find the GCF for the numbers [latex]30[/latex] and [latex]45.[/latex]

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For the numbers [latex]30[/latex] and [latex]45[/latex], the GCF is [latex]15[/latex].

• The factors of [latex]30[/latex] are [latex]1, 2, 3, 5, 6, 10, 15, 30[/latex].
• The factors of [latex]45[/latex] are [latex]1, 3, 5, 9, 15, 45[/latex].

Notice that [latex]15[/latex] is the greatest common factor.

• Find the greatest common factor of [latex]24[/latex] and [latex]36[/latex].

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Step 1: Factor each coefficient into primes. Write all variables with exponents in expanded form.	Factor [latex]24[/latex] and [latex]36[/latex].
Step 2: List all factors–matching common factors in a column.
In each column, circle the common factors.	Circle the [latex]2, 2[/latex], and [latex]3[/latex] that are shared by both numbers.
Step 3: Bring down the common factors that all expressions share.	Bring down the [latex]2, 2, 3[/latex] and then multiply.
Step 4: Multiply the factors.		The GCF of [latex]24[/latex] and [latex]36[/latex] is [latex]12[/latex].

• Find the GCF for the expressions [latex]16x[/latex] and [latex]20x^2[/latex].

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[latex]\begin{align*} \text{Expressions: } & 16x \text{ and } 20x^2 \\ \text{Numerical coefficients: } & 16 \text{ and } 20 \\ \text{Factors of 16: } & 1, 2, 4, 8, 16 \\ \text{Factors of 20: } & 1, 2, 4, 5, 10, 20 \\ \text{Greatest common factor (numerical): } & 4 \\ \text{Variable parts: } & x \text{ in } 16x \text{ (or } x^1\text{)} \text{ and } x^2 \text{ in } 20x^2 \\ \text{Greatest common factor (variables): } & x \\ \text{GCF of expressions: } & 4x \end{align*}[/latex]

• Find the greatest common factor of [latex]14{x}^{3},8{x}^{2},10x[/latex].

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Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column. Bring down the common factors. Multiply the factors.
	The GCF of [latex]14{x}^{3}[/latex] and [latex]8{x}^{2}[/latex] and [latex]10x[/latex] is [latex]2x[/latex]