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# 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) $$a^4 + 2a^2b^2 + b^4$$

(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,

$$= a^4 + a^2b^2 + a^2b^2 + b^4$$

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

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

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