## 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 **x ^{2} − 4** factors as

**(x − 2) (x + 2).**In all cases, a product of simpler objects is obtained.

− Source Wikipedia.

## 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 **x ^{2}, **and in the denominator is

**2x**Take out the common factor, we get,

^{2}. **=> ****`(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 **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))` = 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.**