4.1: Greatest Common Factor
- Page ID
- 173470
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Videos
Lecture: Why We Factor
Example: Factoring the GCF from a Multi-Variable Expression
Example: Factoring Out the GCF Involving a Rational Exponent
Definitions and Theorems
When two or more numbers or expressions are multiplied, each is called a factor of the resulting product.
To factor an expression is to write it as a product of two or more factors.
The greatest common factor (GCF) of two or more integers, not all zero, is the largest positive integer that is a factor of each of them.
The greatest common factor of two or more monomials is the product of the greatest common factor of their coefficients and each variable common to every monomial raised to the smallest exponent with which that variable appears.
If the terms of a sum share a common factor, then the distributive property, applied in reverse, allows that factor to be written outside a set of parentheses:\[ab+ac = a(b+c).\nonumber\]When the shared factor is the greatest common factor of the terms, the remaining expression inside the parentheses has no factor common to all of its terms.
When the leading term of a polynomial is negative, it is customary to factor out the negative of the greatest common factor so that the leading term inside the parentheses is positive. Doing so reverses the sign of every term inside the parentheses.
When a term is itself equal to the greatest common factor, dividing that term by the GCF leaves \(1\), not \(0\). For instance, the GCF of the terms of \(5x^2+5x+5\) is \(5\), and\[5x^2+5x+5 = 5(x^2+x+1),\nonumber\]not \(5(x^2+x)\).
You can check any factorization by multiplying the factors back out using the distributive property. The product must return the original polynomial.
Examples
Find the greatest common factor of \(24\) and \(60\).
- Solution
- Write each integer as a product of primes: \(24 = 2^3\cdot 3\) and \(60 = 2^2\cdot 3\cdot 5\). The primes common to both are \(2\) and \(3\); take each to the smallest power that appears in both, giving \(2^2\cdot 3 = 12\). Thus the greatest common factor is \(12\).
Find the greatest common factor of \(12x^4\), \(18x^3\), and \(30x^2\).
- Solution
- First find the GCF of the coefficients: \(12 = 2^2\cdot 3\), \(18 = 2\cdot 3^2\), and \(30 = 2\cdot 3\cdot 5\), so the common part is \(2\cdot 3 = 6\). The variable \(x\) appears in every monomial; take its smallest exponent, \(x^2\). Hence the greatest common factor is \(6x^2\).
Factor \(6x^2 + 9x\).
- Solution
- The GCF of \(6x^2\) and \(9x\) is \(3x\) (coefficient GCF \(3\); smallest power of \(x\) is \(x\)). Dividing each term by \(3x\) gives \(2x\) and \(3\). Applying the distributive property in reverse,\[6x^2+9x = 3x(2x+3).\nonumber\]
Factor \(15x^3 - 25x^2 + 10x\).
- Solution
- The GCF of the coefficients \(15\), \(25\), and \(10\) is \(5\), and the smallest power of \(x\) present is \(x\), so the GCF is \(5x\). Dividing each term by \(5x\) gives \(3x^2\), \(-5x\), and \(2\). Therefore,\[15x^3-25x^2+10x = 5x(3x^2-5x+2).\nonumber\]
Factor \(-4x^3 - 8x^2 + 12x\), taking the leading coefficient inside the parentheses to be positive.
- Solution
- The GCF of the terms is \(4x\). Because the leading term is negative, factor out \(-4x\). Dividing each term by \(-4x\) reverses every sign: \(-4x^3\div(-4x)=x^2\), \(-8x^2\div(-4x)=2x\), and \(12x\div(-4x)=-3\). Thus,\[-4x^3-8x^2+12x = -4x(x^2+2x-3).\nonumber\]
Factor \(8x^2y - 12xy^2\).
- Solution
- The GCF of the coefficients \(8\) and \(12\) is \(4\). Both terms contain \(x\) (smallest power \(x\)) and \(y\) (smallest power \(y\)), so the variable part of the GCF is \(xy\), and the GCF is \(4xy\). Dividing each term by \(4xy\) gives \(2x\) and \(-3y\). Therefore,\[8x^2y-12xy^2 = 4xy(2x-3y).\nonumber\]
Factor \(x(x-5) + 3(x-5)\).
- Solution
- Both terms share the common factor \((x-5)\). Treating \((x-5)\) as a single quantity and factoring it out,\[x(x-5)+3(x-5) = (x-5)(x+3).\nonumber\]
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Sources
Several parts of this text use modifications from the following sources:
- Wikipedia article: "Factorization"
- Wikipedia article: "Greatest common divisor"
All of these sources are released under the Creative Commons Attribution-Share-Alike License 4.0.