**Factoring Polynomials Examples :**

Here we are going to see some example problems on factoring polynomials.

**Question 1 :**

Factorise the following expressions:

(i) 2a^{2} + 4a^{2} b + 8a^{2} c

**Solution :**

= 2a^{2} + 4a^{2} b + 8a^{2} c

By factoring 2a from the three terms, we get

= 2a (a + 2ab + 4ac)

(ii) ab - ac - mb + mc

**Solution :**

= ab - ac - mb + mc

= a(b - c) - m(b - c)

= (a - m) (b - c)

**Question 2 :**

Factorise the following:

(i) x^{2} + 4x + 4

**Solution :**

= x^{2} + 4x + 4

= x^{2} + 2 ⋅ x ⋅ 2 + 2^{2}

= (x + 2)^{2}

(ii) 3a^{2} - 24ab + 48b^{2}

**Solution :**

= 3a^{2} - 24ab + 48b^{2}

= 3 (a^{2} - 8ab + 16b^{2})

= 3 [a^{2} - 2a(4b) + (4b)^{2}]

= 3 (a - 4b)^{2}

(iii) x^{5} - 16x

**Solution :**

= x^{5} - 16x

= x(x^{4} - 16)

= x [(x^{2})^{2} - 4^{2}]

= x(x^{2} + 4)(x^{2} - 4)

= x(x^{2} + 4)(x + 2)(x - 2)

(iv) m^{2} + 1/m^{2} - 23

**Solution :**

m^{2} + 1/m^{2} - 23 = m^{2} + 1/m^{2} - 25 + 2

= (m^{2} + 1/m^{2} + 2) - 25

= (m + (1/m))^{2 }- 5^{2}

= (m + (1/m) + 5) (m + (1/m) - 5)

(v) 6 - 216 x^{2}

**Solution :**

= 6 - 216 x^{2}

= 6(1 - 36x^{2})

= 6[1 - (6x)^{2}]

= 6(1 + 6x)(1 - 6x)

(vi) a^{2} + 1/a^{2} - 18

**Solution :**

a^{2} + 1/a^{2} - 18 = a^{2} + 1/a^{2} - 16 - 2

= (a^{2} + 1/a^{2} - 2) - 16

= (a - (1/a))^{2 }- 4^{2}

= (a + (1/a) + 4) (a + (1/a) - 4)

**Question 3 :**

Factorise the following:

(i) 4x^{2} + 9y^{2} + 25z^{2} + 12xy + 30yz + 20xz

**Solution :**

= 4x^{2} + 9y^{2} + 25z^{2} + 12xy + 30yz + 20xz

= (2x)^{2}+(3y)^{2}+(5z)^{2}+2(2x)(3y)+2(3y) (5z)+2(5z)(2x)

= (2x + 3y + 5z)^{2}

(ii) 25x^{2} + 4y^{2} + 9z^{2} - 20xy + 12yz - 30zx

**Solution :**

= 25x^{2} + 4y^{2} + 9z^{2} - 20xy + 12yz - 30zx

= (5x)^{2}+(-2y)^{2}+(-3z)^{2}+2(5x)(-2y)+2(-2y)(3z)+2(-3z)(5x)

= (5x - 2y - 3z)^{2}

After having gone through the stuff given above, we hope that the students would have understood, "Factoring Polynomials Examples"

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