Factoring out the greatest common factor
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Factor 12x^3 + 18x^2 completely.
Common mix-up
In this example, a student wrote 12x^3 + 18x^2 = 2x^2(6x + 9).
The student pulled out 2x^2 and wrote 2x^2(6x + 9). It multiplies back correctly, but 6x + 9 still shares a factor of 3, so it is not fully factored.
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See why this works
What is the greatest common factor of 12x^3 and 18x^2?
When every term of a polynomial shares a factor, you can write the whole thing as a product. The tricky part is knowing exactly how much to pull out.
Pull out the largest shared number and the lowest power of each variable that appears in every term.
Factoring means writing an expression as a product. For 12x^3 + 18x^2, whatever you pull out must divide every term evenly. The numbers 12 and 18 share the divisors 1, 2, 3, and 6, and 6 is the largest of them. So the number in the common factor is 6, not 2 or 3, because 6 still divides both 12 and 18.
The variable part works the same way. x^3 means x times x times x, and x^2 means x times x. A factor that is pulled out must fit inside every term, so the exponent in the GCF is the smallest exponent that appears anywhere. Since x^2 appears in both terms, the GCF takes x^2. That is why the common factor here is 6x^2.
Dividing the first term, 12x^3 divided by 6x^2 leaves 2x. Dividing the second term, 18x^2 divided by 6x^2 leaves 3. The factored form is 6x^2(2x + 3), and multiplying back recovers the exact original expression. If you had pulled out only 2x^2, the inside would be 6x + 9, which still shares a factor of 3, so the factoring would be incomplete.
A factored form whose inside still has a common factor is the giveaway that the GCF was too small. Always check by multiplying back: 6x^2 times 2x is 12x^3, and 6x^2 times 3 is 18x^2, which returns to the start. The same recipe works for any polynomial, no matter how many terms it has.
Keep this idea
The GCF is the largest factor shared by every term. Divide each term by it, and nothing shared may remain inside.
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