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Math101learn.math101.caGreatest Common Factor Factoring
Greatest common factor factoring extracts the largest monomial dividing every term of a polynomial. It reverses distribution: $ab+ac=a(b+c)$.
GCF factoring is the first step of almost every polynomial factorization and supports simplification, equation solving, and structural interpretation.
Intuition and core definition
Greatest common factor factoring extracts the largest monomial dividing every term of a polynomial. It reverses distribution: $ab+ac=a(b+c)$. The numerical part is the GCF of coefficients, and each variable uses the smallest exponent present in every term.
Notation, language, and conditions
For coefficients with mixed signs, the GCF is normally positive, but factoring out a negative can make the leading term inside positive. A variable absent from one term has exponent zero, so it is not common. Factoring $0$ or the zero polynomial requires special care; the standard method assumes nonzero terms.
Why this idea matters
Extracting the greatest common factor rewrites a polynomial as a product while leaving the remaining polynomial primitive over the chosen coefficient set.
A dependable method
- Find the numerical GCF of all coefficients.
- For each variable present in every term, choose the minimum exponent.
- Choose the sign of the common factor deliberately.
- Divide each original term by the GCF to build the bracket.
- Distribute the factor back and confirm every term.
Worked example
Representations and interpretation
Exponent columns make the minimum rule visible: list each term’s coefficient factorization and variable exponents, then take only what every row contains. An algebra-tile rectangle represents the GCF as one side length.
Reasoning about variations
After removing the GCF, another pattern may remain. For $3x^3-12x$, first factor $3x$ to obtain $3x(x^2-4)$, then use difference of squares for complete factorization.
Common mistakes
How to check your work
- Divide each original term by the proposed GCF and inspect the quotient.
- Expand the factored form.
- Confirm the bracket terms have GCF $1$ after a greatest factor is removed.
Practice
- Factor completely by taking out the greatest common factor: $24x^3-36x^2$.
- Factor $-8a^2b+20ab^2$.
- Factor $6x^2+9x$ completely.
Answers and brief solutions
Show answers
- $12x^2(2x-3)$ The coefficient GCF is $12$ and minimum $x$ exponent is $2$.
- $-4ab(2a-5b)$ Taking a negative GCF makes the first bracket term positive.
- $3x(2x+3)$ $3x$ divides both terms and leaves relatively prime bracket terms.
Synthesis and transfer
For a collection of rectangular areas sharing both a numeric scale and powers of $x$, the common dimension uses the coefficient GCF and the smallest exponent present.
For $30x^4y^2-45x^3y^5$, the numeric common factor is $15$, while the smallest shared powers are $x^3$ and $y^2$. Factoring produces $15x^3y^2(2x-3y^3)$. The bracket's terms now share no integer or variable factor, confirming that the extraction was greatest. Choosing $5x^2y$ would still yield an equivalent product, but it would leave a removable common factor inside and would not satisfy a request to factor completely by GCF. Division term by term and a final expansion provide complementary checks on the coefficient, exponent, and sign decisions.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Factor completely by taking out the greatest common factor: $24x^3-36x^2$.
- The coefficient GCF is $12$ and minimum $x$ exponent is $2$.
End of lesson
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