Factoring a Numerical GCF

Factor: $8x + 12$

Find the GCF of the coefficients: Factors of 8: 1, 2, 4, 8\nFactors of 12: 1, 2, 3, 4, 6, 12\nGCF = 4 = GCF of numbers is $4$

Check for common variables: First term: $8x$ has $x$\nSecond term: $12$ has no $x$ = No common variable factor

The GCF is just the number: GCF = $4$ = $4$

Divide each term by the GCF: $8x \div 4 = 2x$\n$12 \div 4 = 3$ = $(2x + 3)$

Write the factored form: $8x + 12 = 4(2x + 3)$ = $4(2x + 3)$

Answer: $4(2x + 3)$

Factoring with Variable GCF

Factor: $6x^2 + 9x$

Find the GCF of the coefficients: GCF of 6 and 9 is 3 = Numerical GCF = $3$

Find the GCF of the variables: $x^2$ and $x$\nLowest power of $x$ is $x^1$ = Variable GCF = $x$

Combine for total GCF: $3 \times x = 3x$ = GCF = $3x$

Divide each term by $3x$: $6x^2 \div 3x = 2x$\n$9x \div 3x = 3$ = $(2x + 3)$

Write the factored form: $6x^2 + 9x = 3x(2x + 3)$ = $3x(2x + 3)$

Answer: $3x(2x + 3)$

Factoring with Multiple Variables

Factor: $12x^3y^2 + 18x^2y^3 - 6xy$

Find the GCF of coefficients: GCF of 12, 18, and 6 is 6 = Numerical GCF = $6$

Find the GCF of $x$ terms: $x^3$, $x^2$, $x^1$\nLowest power is $x^1$ = $x$

Find the GCF of $y$ terms: $y^2$, $y^3$, $y^1$\nLowest power is $y^1$ = $y$

Combine for total GCF: $6 \times x \times y = 6xy$ = GCF = $6xy$

Divide each term by $6xy$: $12x^3y^2 \div 6xy = 2x^2y$\n$18x^2y^3 \div 6xy = 3xy^2$\n$-6xy \div 6xy = -1$ = $(2x^2y + 3xy^2 - 1)$

Write the factored form: $12x^3y^2 + 18x^2y^3 - 6xy = 6xy(2x^2y + 3xy^2 - 1)$ = $6xy(2x^2y + 3xy^2 - 1)$

Answer: $6xy(2x^2y + 3xy^2 - 1)$

Factoring a Negative GCF

Factor out $-1$ from: $-x^2 + 5x - 6$

Identify the leading term: The first term is $-x^2$ (negative) = Leading coefficient is negative

Factor out $-1$: Divide each term by $-1$:\n$-x^2 \div (-1) = x^2$\n$5x \div (-1) = -5x$\n$-6 \div (-1) = 6$ = $(x^2 - 5x + 6)$

Write the factored form: $-x^2 + 5x - 6 = -1(x^2 - 5x + 6)$ = $-(x^2 - 5x + 6)$

Answer: $-(x^2 - 5x + 6)$

Mistake: Forgetting to include the variable in the GCF

Why: Students often find the GCF of just the numbers and forget to check for common variable factors.

Correct: Always check BOTH numbers AND variables. For $6x^2 + 9x$, the GCF is $3x$, not just $3$.

Mistake: Using the highest power instead of lowest

Why: For variables, students sometimes take the highest exponent instead of the lowest.

Correct: The GCF uses the LOWEST power. For $x^3$ and $x^2$, use $x^2$ (what goes into both).

Mistake: Not dividing all terms

Why: Students sometimes forget to divide one of the terms by the GCF.

Correct: Check your work by distributing: $3x(2x + 3) = 6x^2 + 9x$ confirms it's correct.

Mistake: Stopping when GCF is 1

Why: If terms share no common factor, students may think they made an error.

Correct: Some polynomials have GCF = 1. Example: $x^2 + 3x + 5$ cannot be factored by GCF.

Simplifying Area Formulas

Architects and engineers use factoring to simplify expressions for areas and volumes.

If a rectangular room has area $12x^2 + 18x$ square meters, factoring gives $6x(2x + 3)$, showing the dimensions.

Analyzing Profit Functions

Business analysts factor expressions to understand profit relationships.

If profit is $P = 100x - 20x^2$, factoring gives $P = 20x(5 - x)$, showing profit is zero when $x = 0$ or $x = 5$.

Q: What if there is no common factor?

A: If the GCF is 1, you cannot factor out a GCF. Move on to other factoring methods like grouping or using special patterns.

Q: Should I factor out a negative GCF?

A: Sometimes yes! If the leading term is negative, factoring out $-1$ (or a negative GCF) can make the remaining expression easier to work with.

Q: How do I know I found the right GCF?

A: Distribute your answer back out. If you get the original expression, you factored correctly. Also, check that the terms inside the parentheses have no remaining common factor.