**Types Of Factorization Example Problems With Solutions**

**Type I: Factorization by taking out the common factors. **

**Example 1: **Factorize the following expression

2x^{2}y + 6xy^{2} + 10x^{2}y^{2}

**Solution: **2x^{2}y + 6xy^{2} + 10x^{2}y^{2}

=2xy(x + 3y + 5xy)

**Type II: Factorization by grouping the terms. **

**Example 2: **Factorize the following expression

a^{2} – b + ab – a

**Solution: **a^{2} – b + ab – a

= a^{2} + ab – b – a = (a^{2} + ab) – (b + a)

= a (a + b) – (a + b) = (a + b) (a – 1)

**Type III: Factorization by making a perfect square. **

**Example 3: **Factorize of the following expression

9x^{2} + 12xy + 4y^{2}

**Solution: **9x^{2} + 12xy + 4y^{2}

= (3x)^{2} + 2 × (3x) × (2y) + (2y)^{2}

= (3x + 2y)^{2}

**Example 4: **Factorize of the following expression

\(\frac{{{x}^{2}}}{{{y}^{2}}}+2+\frac{{{y}^{2}}}{{{x}^{2\prime }}},x\ne 0,y\ne 0\)

**Solution: **

**Example 5: **Factorize of the following expression

\({{\left( 5x-\frac{1}{x} \right)}^{2}}+4\left( 5x-\frac{1}{x} \right)+4,x\ne 0\)

**Solution:**

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**Type IV: Factorizing by difference of two squares.**

**Example 6: ** Factorize the following expressions

(a) 2x^{2}y + 6 xy^{2} + 10 x^{2}y^{2}

(b) 2x^{4} + 2x^{3}y + 3xy^{2} + 3y^{3}

**Solution:**

**Example 7: **Factorize 4x^{2} + 12 xy + 9 y^{2}

**Solution:**

**Example 8: **Factorize each of the following expressions

(i) 9x^{2} – 4y^{2}

(ii) x^{3} – x

**Solution:**

**Example 9: **Factorize each of the following expressions

(i) 36x^{2} – 12x + 1 – 25y^{2}

\(\text{(ii) }{{a}^{2}}-\frac{9}{{{a}^{2}}},a\ne 0\)

**Solution:**

**Example 10: **Factorize the following algebraic expression

x^{4} – 81y^{4}

**Solution:**

**Example 11: **Factorize the following expression

x(x+z) – y (y+z)

**Solution: **x(x+z) – y (y+z) = (x^{2} – y^{2}) + (xz–yz)

= (x–y) (x+y) + z (x–y)

= (x–y) {(x+y) + z}

= (x–y) (x+ y + z)

**Example 12: **Factorize the following expression

x^{4} + x^{2} + 1

**Solution: **x^{4} + x^{2} + 1 = (x^{4} + 2x^{2} +1) – x^{2}

= (x^{2} +1)^{2} – x^{2} = (x^{2} + 1 – x) (x^{2} + 1+x)

= (x^{2}–x + 1) (x^{2} + x + 1)

**Type V: Factorizing the sum and difference of cubes of two quantities.**

(i) (a^{3} + b^{3}) = (a + b) (a^{2} – ab + b^{2})

(ii) (a^{3} – b^{3}) = (a – b) (a^{2} + ab + b^{2})

**Example 13: **Factorize the following expression

a^{3} + 27

**Solution: **a^{3} + 27 = a^{3} + 3^{3} = (a + 3) (a^{2} –3a +9)

**Example 14: **Simplify : (x+ y)^{3} – (x –y)^{3} – 6y(x^{2} – y^{2})

**Solution: **