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Factoring Expressions Study Guide (page 2)

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Updated on Oct 3, 2011

Example

Factor 18x3 + 12x.

First, find the largest whole number that divides evenly into both coefficients. The factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 12 are 1, 2, 3, 4, 6, and 12. The greatest common factor is 6. We can factor out a 6 from both terms. Next, look at the variables. Both terms have x in common. The lowest exponent of x is 1, in the second term. We can factor out x from both terms.

To factor 6x out of both terms, divide both terms by 6x: 18x3 ÷ 6x = 3x2 and 12x ÷ 6x = 2. 8x3 + 12x = 6x(3x2 + 2). We can check our work by multiplying: 6x(3x2 + 2) = (6x)(3x2) + 6x)(2) = 18x3 + 12x, so we have factored correctly

Some expressions can only have a whole number factored out, and other expressions can only have a variable factored out. Some expressions, such as 10x + 7, cannot be factored at all. There is no whole number (other than 1) that divides evenly into 10 and 7, and there is no variable common to both terms.

Factoring Multivariable Expressions

Just as with single–variable expressions, a coefficient or variable can only be factored out of a multivariable expression if the coefficient or variable appears in every term. If two variables are common to every term, then two variables can be factored out of the expression.

Example

Factor 9x5y2 + 12xy3.

Start with the coefficients. The greatest common factor of 9 and 12 is 3, so 3 can be factored out of the expression.

Next, check to see if each variable exists in each term: x is in both terms and so is y. Both can be factored out of the expression. The smallest exponent of x is 1 (in 12xy3) and the smallest exponent of y is 2 (in 9x5y2).

Divide each term by 3xy2: 9x5y2 ÷3xy2 = 3x4, and 12xy3 ÷3xy2 = 4y.

9x5y2 + 12xy3 = 3xy2(3x4 + 4y)

Example

Factor 7x6y6–2x5 + 8x5y4.

The greatest common factor of 7, 2, and 8 is 1, so no constant can be factored out of the expression. The variable x is common to every term, so it can be factored out, but the variable y is not in the middle term, –2x5, so it cannot be factored out. The smallest exponent of x is 5, in both the second and third terms, so we can factor x5 out of every term Divide each term by x5: 7x6y6 ÷ x5 = 7xy6, 2x5 ÷ x5 = 2, and 8x5y4 ÷x5 = 8y4. 7x6y6– 2x5 + 8x5y4 =x5(7xy6– 2 + 8y4)

Example

Factor 16a4b9c4– 24a3b12c6 + 40a5b7c5.

The greatest common factor of the coefficients, 16, 24, and 40, is 8. The variables a, b, and c are common to every term. The smallest exponent of a is 3, the smallest exponent of b is 7, and the smallest exponent of c is 4, so 8a3b7c4 can be factored out of every term: 16a4b9c4 ÷ 8a3b7c4 = 2ab2, 24a3b12c6÷ 8a3b7c4 = 3b5c2, and 40a5b7c5 ÷ 8a3b7c4 = 5a2c.

16a4b9c4 – 24a3b12c6 + 40a5b7c5 = 8a3b7c4(2ab2 – 3b5c2 + 5a2c)

Find practice problems and solutions at Factoring Expressions Practice Questions.

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