Example Card: Factoring with a GCF
Factor polynomials using the GCF in Grade 9 algebra: always identify and factor out the greatest common factor first to simplify the remaining expression before applying other factoring techniques.
Key Concepts
Always look for a common factor first; it simplifies the entire puzzle. This example shows how to use the greatest common factor (GCF) to make factoring easier.
Example Problem Factor the expression $3x^3 + 9x^2 30x$ completely.
Step by Step 1. First, find the greatest common factor (GCF) of all the terms in the polynomial. The GCF of $3x^3$, $9x^2$, and $ 30x$ is $3x$. 2. Factor out the GCF from the expression: $$ 3x^3 + 9x^2 30x = 3x(x^2 + 3x 10) $$ 3. Now, focus on the trinomial inside the parentheses, $x^2 + 3x 10$. Find two numbers that have a product of $ 10$ and a sum of $3$. $$ 5 imes ( 2) = 10 ext{ and } 5 + ( 2) = 3 $$ 4. Use these two numbers, $5$ and $ 2$, to write the trinomial as a product of two binomials. $$ 3x(x^2 + 3x 10) = 3x(x + 5)(x 2) $$.
Common Questions
How do you factor a polynomial by first finding the GCF?
Identify the greatest factor dividing all terms — both the largest common numerical factor and the lowest power of any shared variable. Divide each term by this GCF and write it as the product GCF × (remaining polynomial).
How do you factor 4x³ + 8x² - 12x using the GCF?
Find GCF: largest number dividing 4, 8, 12 is 4; lowest power of x is x¹. GCF = 4x. Divide each term: 4x³/4x = x², 8x²/4x = 2x, -12x/4x = -3. Result: 4x(x² + 2x - 3). Further factor: 4x(x+3)(x-1).
Why is factoring out the GCF the essential first step?
Factoring the GCF reduces the size of remaining coefficients and degree, making subsequent factoring (trinomials, difference of squares) significantly easier and less error-prone. It's the first step every time you factor.