Factoring the Greatest Common Factor
Factor polynomials by identifying and extracting the greatest common factor (GCF) from all terms, simplifying expressions in Grade 9 Algebra.
Key Concepts
Property The greatest common factor (GCF) of polynomials is the largest polynomial that divides evenly into the polynomials. To factor out the GCF: 1. Identify the GCF of the coefficients. 2. Identify the GCF of the variables (using the lowest power for each variable). 3. Combine to find the expression's GCF. 4. Determine what the GCF must be multiplied by to get each term. 5. Write the factored expression as the GCF multiplied by the remaining terms.
Examples To factor $8a^2b^3 + 12ab^2 + 20a^3b$, the GCF is $4ab$. Factoring this out gives $4ab(2ab^2 + 3b + 5a^2)$. To factor $15m^4n 25m^2n^2 + 5mn$, the GCF is $5mn$. The factored form is $5mn(3m^3 5mn + 1)$. To factor $12x^2(y 1) + 8x(y 1)$, the GCF is $4x(y 1)$. This results in $4x(y 1)(3x + 2)$.
Explanation Think of this as reverse distribution. You are finding the largest number and variable part that every single term shares and pulling it out to the front. This simplifies the expression inside the parentheses, making it easier to work with.
Common Questions
How do you find the greatest common factor (GCF) of a polynomial?
To find the GCF, list all factors of each coefficient and identify the highest number that divides evenly into all of them. Then find the lowest power of any shared variable. The GCF is the product of the common numerical factor and the lowest common variable powers.
What is the first step in factoring any polynomial?
Always check for a greatest common factor first before using any other factoring method. Factoring out the GCF simplifies the remaining expression, making subsequent factoring (like trinomial factoring or difference of squares) much easier and less error-prone.
How do you factor out the GCF from a polynomial?
Write the GCF outside parentheses, then divide each term of the polynomial by the GCF to find what goes inside the parentheses. For example, 6x² + 9x = 3x(2x + 3) because 3x is the GCF, and 6x²÷3x = 2x and 9x÷3x = 3.