Examples of Factoring

Introduction for Factoring:

In mathematics, factorization or factoring is the decomposition of an object 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 x2 − 4 factors as (x − 2) (x + 2). In all cases, a product of simpler objects is obtained.

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Solved Examples of Factoring:

Example 1: Solve by factoring, `(4x^3 - 9x^2)/(8x^3 - 18x^2)` = 0.

Solution: Given that, `(4x^3 - 9x^2)/(8x^3 + 18x^2)` = 0.

Here the common factor in the numerator is x2, and in the denominator is 2x2. Take out the common factor, we get,

=> `(4x^3 - 9x^2)/(8x^3 + 18x^2) = (x^2(4x -9))/(2x^2(4x+9))`= 0,

=> `(4x - 9)/(2(4x + 9))` = 0.

Cross multiplying the above expression we get the right hand side value as zero,

=> 4x − 9 = 0,

=> 4x = 9,

=> x = `9/4`.

Hence the solution is,  x = `9/4`.

Example 2: Solve by factoring, `(3x^2)/(12x^2 - 6x)` = 0.

Solution: Given that, `(3x^2)/(12x^2 - 6x)` = 0.

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

=> `(3x^2)/(12x^2 -6x) = (3x^2)/(3x(4x-2)) ` = 0,

=> `x/(4x -2)` = 0,

Cross multiplying the above expression we get the right hand side value as zero,

=> x = 0.

Hence the solution is, x = 0.

Example 3: Solve by factoring, `(4x - 8)/(4x^2-4x)` = 0.

Solution: Given that, `(4x - 8)/(4x^2-4x)` = 0.

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))` = 0,

Cross multiplying the above expression we get the right hand side value as zero,

=> x − 2 = 0,

=> x = 2.

Hence the solution is, x = 2.

Example 4: Solve by factoring, `(4x+6y-8z)/(12x -8xy)` = 0.

Solution: Given that, `(4x+6y-8z)/(12x -8xy)` = 0.

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))` = 0,

=> `(2x+3y-4z)/(2x(3-2y))` = 0,

Cross multiplying the above expression we get the right hand side value as zero,

=> 2x + 3y − 4z = 0.

Hence the solution is, 2x + 3y − 4z = 0.

Example 5: Solve by factoring, `(m^2 - 16n^2)/(2m + 8n)` = 0.

Solution: Given that, `(m^2 - 16n^2)/(2m + 8n)` = 0.

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

=>   `((m+4n)(m-4n))/(2(m+4n))` = 0,

=> `(m-4n)/2` = 0,

Cross multiplying the above expression we get the right hand side value as zero,

=> m − 4n = 0,

=> m = 4n.

Hence the solution is, m = 4n.

Example 6: Solve by factoring, `(12x^3 - 12x^2)/(6x^2 + 6x)` = 0.

Solution: Given that, `(12x^3 - 12x^2)/(6x^2 + 6x)` = 0.

Here the common factor in the numerator is 12x2, 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))` = 0.

=> `(x(x-1))/(x+1)` = 0,

Cross multiplying the above expression we get the right hand side value as zero,

=> x(x 1) = 0,

=> x = 0, and x 1 = 0,

=> x = 0, and x = 1.

Hence the solution is, x = 0, and x = 1.

Practice Problems for Factoring:

Example 1: Simplify the algebraic fraction by factoring, `(x^2-25)/(2x + 10)` = 0.

Solution: x = 5.

Example 2: Simplify the algebraic fraction by factoring, `(16x^4 - 9x^2)/(4x^2+3x)` = 0.

Solution: x = 0, and x = `3/4`.

Example 3: Simplify the algebraic fraction by factoring, `(x^2-x-2)/(x^2-2x-3)` = 0.

Solution: x = 2.