## MP Board Class 8th Maths Solutions Chapter 14 Factorization Ex 14.2

Question 1.

Factorise the following expressions.

(i) a^{2} + 8o + 16

(ii) p^{2} – 10p + 25

(iii) 25m^{2} + 30m+ 9

(iv) 49y^{2} + 84yz + 36z^{2}

(v) 4x^{2} – 8x + 4

(vi) 121b^{2} – 88bc + 16c^{2}

(vii) (l + m)^{2} – 4lm (Hint: Expand (l + m)^{2} first)

(viii) a^{4} + 2a^{2}b^{2} + b^{4}

Solution:

(i) The expression is a^{2} + 8a +16

= a^{2} + 2 × 4 × a + (4)^{2}

= (a + 4)^{2} = (a + 4) (a + 4)

(ii) The expression is p^{2} – 10p + 25

= (p)^{2} – 2 × 5 × p + (5)^{2}

= (P – 5)^{2} = (p – 5)(p – 5)

(iii) The expression is 25m^{2} + 30m + 9

= (5m)^{2} + 2 × 3 × (5m) + (3)^{2} = (5m + 3)^{2}

= (5m + 3) (5m + 3)

(iv) The expression is 49y^{2} + 84yz + 36z^{2}

= (7y)^{2} + 2 × (7y) × (6z) + (6z)^{2} = (7y + 6z)^{2}

= (7y + 6z)(7y + 6z)

(v) The expression is 4x^{2} – 8x + 4

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

= [2(x – 1)]^{2} = 4(x – 1)^{2}

= 4(x – 1)(x – 1)

(vi) The expression is 121b^{2} – 88bc + 16c^{2}

= (11b)^{2} – 2 × (11b) (4c) + (4c)^{2} = (11b – 4c)^{2}

= (11b – 4c)(11b – 4c)

(vii) The expression is (l + m)^{2} – 4lm

= l^{2} + 2 × l × m + m^{2} – 4lm [ ∵ (a + b)^{2} = a^{2} + 2ab + b^{2}]

= l^{2} + 2lm + m^{2} – 4lm = l^{2} – 2lm + m^{2}

= (l – m)^{2} = (l – m)(l – m)

(viii)The expression is a^{4} + 2a^{2}b^{2} + b^{4}

= (a^{2})^{2} + 2 × a^{2} × b^{2} + (b^{2})^{2} = (a^{2} + b^{2})^{2}

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

Question 2.

Factorise.

(i) 4p^{2} – 9q^{2}

(ii) 63a^{2} – 112b^{2}

(iii) 49x^{2} – 36

(iv) 16x^{5} – 144x^{3}

(v) (l + m)^{2} – (l – m)^{2}

(vi) 9x^{2}y^{2} – 16

(vii) (x^{2} – 2xy + y^{2}) – z^{2}

(viii)25a^{2} – 4b^{2} + 28bc – 49c^{2}.

Solution:

(i) The expression is 4p^{2} – 9q^{2}

= (2P)^{2} – (3q)^{2} = (2p + 3q) (2p – 3q)

(ii) The expression is 63a^{2} – 112b^{2}

= 7[9a^{2} – 16b^{2}] = 7[(30)^{2} – (4b)^{2}]

= 7(3o + 4b) (3a – 4b).

(iii) The expression is 49x^{2} – 36 = (7x)^{2} – (6)^{2}

= (7x + 6) (7x – 6).

(iv) The expression is 16x^{5} – 144x^{3}

= 16x^{3}(x^{2} – 9) = 16x^{3} (x^{2} – 3^{2})

= 16x^{3} (x + 3)(x – 3).

(v) The expression is (l + m)^{2} – (l – m)^{2}

= (l^{2} + 2lm + m^{2}) – (l^{2} – 2lm + m^{2})

= l^{2} + 2lm + m^{2} – l^{2} + 2lm – m^{2} = 4lm

(vi) The expression is 9x^{2}y^{2} – 16

= (3xy)^{2} – (4)^{2} = (3xy + 4) (3xy – 4)

(vii) The expression is (x^{2} – 2xy + y^{2}) – z^{2}

= (x – y)^{2} – z^{2} = (x – y + z) (x – y – z)

(viii)The expression is 25a^{2} – 4b^{2} + 28bc – 49c^{2}

= 25a^{2} – [4b^{2} – 28bc + 49c^{2}]

= 25a^{2} – [(2b)^{2} – 2 × (2b) × (7c) + (7c)^{2}]

= (5a)^{2} – (2b – 7c)^{2}

= (5a + 2b – 7c) (5a – 2b + 7c)

Question 3.

Factorise the expressions.

(i) ax^{2} + bx

(ii) 7p^{2} + 21q^{2}

(iii) 2x^{3} + 2xy^{2} + 2xz^{2}

(iv) am^{2} + bm^{2} + bn^{2} + an^{2}

(v) (lm + l) + m + 1

(vi) y(y + z) + 9(y + z)

(vii) 5y^{2} – 20y – 8z + 2yz

(viii) 10ab + 4a + 5b + 2

(ix) 6xy – 4y + 6 – 9x

Solution:

(i) The expression is ax^{2} + bx = x(ax + b)

(ii) The expression is 7p^{2} + 21q^{2} = 7(p^{2} + 3q^{2} )

(iii) The expression is 2x^{3} + 2xy^{2} + 2xz^{2}

= 2x(x^{2} + y^{2} + z^{2} ).

(iv) The expression is am^{2} + bm^{2} + bn^{2} + an^{2}

= m^{2} (a + b) + n^{2} (b + a) = (m^{2} + n^{2} ) (a + b)

(v) The expression is (lm + l) + m + 1

= l(m + 1) + 1 (m + 1)= (l + 1) (m + 1)

(vi) The expression is y(y + z) + 9(y + z)

= (y + 9) (y + z).

(vii) The expression is 5y^{2} – 20y – 8z + 2yz

= 5y(y – 4) + 2z(y – 4) = (5y + 2z)(y – 4)

(viii) The expression is 10ab + 4a + 5b + 2

= 2a(5b + 2) + 1 (5b + 2) = (2a + 1) (5b + 2)

(ix) The expression is 6xy – 4y + 6 – 9x

= 2y(3x – 2) – 3 (3x – 2) = (2y – 3) (3x – 2)

Question 4.

Factorise.

(i) a^{4} – b^{4}

(ii) p^{4} – 81

(iii) x^{4} – (y + z)^{4}

(iv) x^{4} – (x – z)^{4}

(v) a^{4} – 2a^{2}b^{2} + b^{4}

Solution:

(i) The expression is a^{4} – b^{2}

= (a^{2})^{2} – (b^{2})^{2}

= (a^{2} + b^{2}) (a^{2} – b^{2})

= (a^{2} + b^{2})(a + b)(a – b).

(ii) The expression is p^{4} – 81 = (p)^{4} – (3)^{4}

= (P^{2})^{2} – (3^{2})^{2} = (p^{2} + 3^{2}) (p^{2} – 3^{2})

= (p^{2} + 9) (p + 3) (p – 3)

(iii) The expression is x^{4} – (y + z)^{4}

= (x^{2})^{2} – ((y + z)^{2})^{2}

= [x^{2} + (y + z)^{2}] [x^{2} – (y + z)^{2}]

= [x^{2} + (y + z)^{2}] (x + y + z) (x – (y + z))

= [x^{2} + (y + z)^{2}] (x + y + z) (x – y – z)

(iv) The expression is x^{4} – (x – z)^{4}

= (x^{2})^{2} – ((x – z)^{2})^{2}

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

= (x – x + z)(x + x – z)(x^{2} + x^{2} + z^{2} – 2xz)

= z(2x – z) (2x^{2} – 2xz + z^{2}).

(v) The expression is a^{4} – 2a^{2}b^{2} + b^{4}

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

= [(a + b) (a – b)]^{2} = (a + b)^{2} (a – b)^{2}

Question 5.

Factorise the following expressions.

(i) p^{2} + 6p + 8

(ii) q^{2} – 10q + 21

(iii) p^{2} + 6p – 16

Solution:

(i) The expression is p^{2} + 6p + 8

= p^{2} + 4p + 2p + 8 = p(p + 4) + 2 (p + 4)

= (p + 2) (p + 4)

(ii) The expression is q^{2} – 10q + 21

= q^{2} – 7q – 3q + 21 = q(q – 7) – 3 (q – 7)

= (q – 3) (q – 7)

(iii) The expression is p^{2} + 6p – 16

= p^{2} + 8p – 2p – 16

= p(p + 8) – 2(p + 8) = (p – 2) (p + 8)