Add the following algebraic expressions:
(i) $2 x^{2} + 3 x + 5, 3 x^{2} - 4 x - 7$
(ii) $x^{2} - 2 x - 3, x^{2} + 3 x + 1$
(iii) $2 t^{2} + t - 4, 1 - 3 t - 5 t^{2}$
(iv) $x y - y z, y z - x z, z x - x y$
(v) $a^{2} + b^{2}, b^{2} + c^{2}, c^{2} + a^{2}, 2 a b + 2 b c + 2 c a$

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Solution

We need to perform addition on the following algebraic expressions:

(i) Given, $2 x^{2} + 3 x + 5$ and $3 x^{2} - 4 x - 7$
Thus we have $(2 x^{2} + 3 x + 5) + (3 x^{2} - 4 x - 7) = 5 x^{2} - x - 2$

(ii) Given, $x^{2} - 2 x - 3$ and $x^{2} + 3 x + 1$
Thus we have $(x^{2} - 2 x - 3) + (x^{2} + 3 x + 1) = 2 x^{2} + x - 2$

(iii) Given, $2 t^{2} + t - 4$ and $1 - 3 t - 5 t^{2}$
Thus we have $(2 t^{2} + t - 4) + (1 - 3 t - 5 t^{2}) = - 3 t^{2} - 2 t - 3$

(iv) Given, $x y - y z, y z - x z$ and $x z - x y$
Thus we have $(x y - y z) + (y z - x z) + (z x - x y) = 0;$

(v) Given, $a^{2} + b^{2}, b^{2} + c^{2}, c^{2} + a^{2}$ and $2 a b + 2 b c + 2 c a$
Thus we have $(a^{2} + b^{2}) + (b^{2} + c^{2}) + (c^{2} + a^{2}) + (2 a b + 2 b c + 2 a c) = (a + b)^{2} + (b + c)^{2} + (c + a)^{2}$

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