How to Factor Out the Greatest Common Factor

Step-by-step solution

1. Find the greatest common factor of the coefficients.

   $$\gcd(6, 9) = 3$$

   Why: Both coefficients divide by 3, so 3 comes outside the bracket.

2. Find the common variable factor.

   $$x^2 = x \cdot x,\quad x = x \cdot 1 \Rightarrow \text{common factor } x$$

   Why: The smallest power of x in both terms is x.

3. Divide each term by 3x and write the factored form.

   $$6x^2 \div 3x = 2x,\quad 9x \div 3x = 3 \Rightarrow 3x(2x + 3)$$

   Why: Each quotient becomes one term inside the bracket.

4. Check by expanding.

   $$3x(2x + 3) = 6x^2 + 9x$$

   Why: Expanding returns the original expression, so the factor is correct.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

1. 1

   $4x^2 + 8x$

   Show answer

   $4x(x + 2)$

2. 2

   $5x^3 - 10x^2$

   Show answer

   $5x^2(x - 2)$

3. 3

   $12x^2y + 18xy^2$

   Show answer

   $6xy(2x + 3y)$

4. 4

   $7x^2 - 14x$

   Show answer

   $7x(x - 2)$

5. 5

   $9x^3 + 3x^2$

   Show answer

   $3x^2(3x + 1)$