1. Factorise the following expressions.
(i) \(a^2 + 8a +16\)

(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) \(a^4 + 2a^2b^2 + b^4\)


Answer :

(i) Given:\(a^2 + 8a + 16\)

Here, 4 + 4 = 8 and \(4 \times 4 = 16\)

\(a^2 + 8a +16\)

\(= a^2 + 4a + 4a + 4 \times 4\)

\(= (a^2 + 4a) + (4a + 16)\)

\(= a(a + 4) + 4(a + 4)\)

\(= (a + 4) (a + 4)\)

\(= (a + 4)^2\)

(ii) Given:\(p^2 – 10p + 25\)

Here, 5 + 5 = 10 and \(5 \times 5 = 25\)

\(=p^2 – 10p + 25\)

\(= p^2 – 5p – 5p + 5 \times 5\)

\(= (p^2 – 5p) + (-5p + 25)\)

\(= p(p – 5) – 5(p – 5)\)

\(= (p – 5) (p – 5)\)

\(= (p – 5)^2\)

(iii) Given:\(25m^2 + 30m + 9\)

Here, \(15 + 15 = 30\) and \(15 \times 15 = 25 \times 9 = 225\)

\(25m^2 + 30m + 9\)

\(= 25m^2 + 15m + 15m + 9\)

\(= (25m^2 + 15m) + (15m + 9)\)

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

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

\(= (5m + 3)^2\)

(iv) Given:\(49y^2 + 84yz + 36z^2\)

Here, \(42 + 42 = 84\) and \(42 \times 42 = 49 \times 36 = 1764\)

\(49y^2 + 84yz + 36z^2\)

\(= 49y^2 + 42yz + 42yz + 36z^2\)

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

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

\(= (7y + 6z)^2\)

(v) Given:\(4x^2 – 8x + 4\)

\(= 4(x^2 – 2x + 1) [Taking \;4 \;common]\)

\(= 4(x^2 – x – x + 1)\)

\(= 4[x(x – 1) -1(x – 1)]\)

\(= 4(x – 1)(x – 1)\)

\(= 4(x – 1)^2\)

(vi) Given: \(121b^2 – 88bc + 16c^2\)

Here, \(44 + 44 = 88\) and \(44 \times 44 = 121 \times 16 = 1936\)

\(121b^2 – 88bc + 16c^2\)

\(= 121b^2 – 44bc – 44bc + 16c^2\)

\(= 11b(11b – 4c) – 4c(11b – 4c)\)

\(= (11b – 4c) (11b – 4c)\)

\(= (11b – 4c)^2\)

(vii) Given:\((l + m)^2 – 4lm\)

Expanding the expression, \( (l + m)^2\), we have

\(l^2 + 2lm + m^2 – 4lm\)

\(= l^2 – 2lm + m^2\)

\(= l^2 – Im – lm + m^2\)

\(= l(l – m) – m(l – m)\)

\(= (l – m) (l – m)\)

\(= (l – m)^2\)

(viii) Given:\(a^4 + 2a^2b^2 + b^4\).So we have,

\(= a^4 + a^2b^2 + a^2b^2 + b^4\)

\(= a^2(a^2 + b^2) + b^2(a^2 + b^2)\)

\(= (a^2 + b^2)(a^2 + b^2)\)

\(= (a^2 + b^2)^2\)

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