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subject: Factoring Algebraic Fractions [print this page]

Introduction to Factoring Algebraic Fractions:

In mathematics, factorization (also factorisation in British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 5, and the polynomial x^2 - 4 factors as (x - 2) (x + 2). In all cases, a product of simpler objects is obtained. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra.

- Source Wikipedia.

Examples of Factoring Algebraic Fractions:

Example 1: Simplify the algebraic fraction by factoring, '(4x^3 - 9x^2)/(8x^3 - 18x^2)'.

Solution: Given that, '(4x^3 - 9x^2)/(8x^3 + 18x^2)'.

Here the common factor in the numerator is x^2, and in the denominator is 2x^2. Take out the common factor, we get,

=> '(4x^3 - 9x^2)/(8x^3 + 18x^2)=(x^2(4x-9))/(2x^2(4x + 9))',

=> '(4x - 9)/(2(4x + 9))'.

Hence the solution is, '(4x - 9)/(2(4x + 9))'.

Example 2: Simplify the algebraic fraction by factoring, '(3x)/(12x^2 - 6x)'.

Solution: Given that, '(3x)/(12x^2 - 6x)'.

Here the common factor is x, and 3. Hence by take out the common factor, we get,

=> '(3x)/(12x^2 -6x) = (3x)/(3x(4x-2)) ',

=> '1/(4x -2)'.

Hence the solution is, '1/(4x -2)'.

Example 3: Simplify the algebraic fraction by factoring, '(4x - 8)/(4x^2-4x)'.

Solution: Given that, '(4x - 8)/(4x^2-4x)'.

Here the common factor in the numerator is 4, and in the denominator is 4x. Take out the common factor, we get,

=> '(4x- 8)/(4x^2-4x)= (4(x-2))/(4x(x-1))',

=> '(x-2)/(x(x-1))'.

Hence the solution is, '(x-2)/(x(x-1))'.

Example 4: Simplify the algebraic fraction by factoring, '(4x+6y-8z)/(12x -8xy)'.

Solution: Given that, '(4x+6y-8z)/(12x -8xy)'.

Here the common factor in the numerator is 2, and in the denominator is 4x. Take out the common factor, we get,

=> '(4x+6y-8z)/(12x -8xy)=(2(2x+3y-4z))/(4x(3-2y))',

=> '(2x+3y-4z)/(2x(3-2y))'.

Hence the solution is, '(2x+3y-4z)/(2x(3-2y))'.

Example 5: Simplify the algebraic fraction by factoring, '(2m+4n)/(m^2 - 16n^2)'.

Solution: Given that, '(2m+4n)/(m^2 - 16n^2)'.

Here the common factor in the numerator is 2, and factoring the denominator we get, (m + 4n) (m - 4n).

=> '(2(m+4n))/((m+4n)(m-4n))',

=> '2/(m-4n)'.

Hence the solution is, '2/(m-4n)'.

Example 6: Simplify the algebraic fraction by factoring, '(12x^3 - 12x^2)/(6x^2 + 6x)'.

Solution: Given that, '(12x^3 - 12x^2)/(6x^2 + 6x)'.

Here the common factor in the numerator is 12x^2, and in the denominator is 6x. Take out the common factor, we get,

=> '(12x^3 - 12x^2)/(6x^2 - 6x) = (12x^2(x-1))/(6x(x+1))',

=> '(x(x-1))/(x+1)'.

Hence the solution is, '(x(x-1))/(x+1)'.

Practice Problem for Factoring Algebraic Fractions:

Example 1: Simplify the algebraic fraction by factoring, '(x+5)/(2x + 10)'.

Solution: '1/2'.

Example 2: Simplify the algebraic fraction by factoring, '(4x^2 - 9y^2)/(2x-3y)'.

Solution: 2x + 3y.

Example 3: Simplify the algebraic fraction by factoring, '(x^2-x-2)/(x^2-2x-3)'.

Solution: '(x-2)/(x-3)' .

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by: mathmaster

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