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# Verify whether the following are zeroes of the polynomial, indicated against them. i)$$p(x) = 3x + 1, x = \frac{-1}{3}$$ ii)$$p(x) = 5x - \pi , x = \frac{4}{5}$$ iii)$$p(x) = x^2 - 1$$, x = 1, -1iv)$$p(x) = (x + 1)(x - 2)$$, x = -1, 2v)$$p(x) = x^2$$, x = 0vi)$$p(x) = lx + m, x = \frac{-m}{l}$$vii)$$p(x) = 3x^2 - 1$$, $$x = \frac{-1}{\sqrt{3}}$$, $$\frac{2}{\sqrt{3}}$$viii)$$p(x) = 2x + 1, x = \frac{1}{2}$$

i) $$p(x) = 3x + 1$$

To check, if $$x = \frac{-1}{3}$$ is zero of given polynomial then $$p(\frac{-1}{3})$$ should be equal to 0.

$$\therefore p(\frac{-1}{3})$$
$$\Rightarrow p(\frac{-1}{3} ) = 3(\frac{-1}{3} ) + 1$$
$$\Rightarrow p(\frac{-1}{3})= -1 +1 = 0$$

$$\therefore$$ $$x = \frac{-1}{3}$$ is zero of given polynomial.

ii) $$p(x) = 5x - \pi$$

To check, if $$\frac{4}{5}$$is zero of given polynomial then $$p(\frac{4}{5} )$$ should be equal to 0.

$$\therefore p(\frac{4}{5} ) = 5(\frac{4}{5} ) - \pi$$
$$\Rightarrow p(\frac{4}{5} ) = 4 - \pi$$

$$\therefore x = \frac{4}{5}$$ is not a zero of given polynomial.

iii) $$p(x) = x^2 - 1$$

To check, if x = 1 is zero of given polynomial then p(1) should be equal to 0.

$$\therefore p(1) = 1^2 - 1$$
$$\Rightarrow p(1) = 1 - 1 = 0$$

$$\therefore x = 1$$ is a zero of given polynomial.

To check, if x = -1 is zero of given polynomial then p(-1) should be equal to 0.

$$\therefore p(-1) = {-1}^2 - 1$$
$$\Rightarrow p(-1) = 1 - 1 = 0$$

$$\therefore, x = 1$$ is too, a zero of given polynomial.

iv) $$p(x) = (x + 1)(x - 2)$$

To check, if x = -1 is zero of given polynomial then p(-1) should be equal to 0.

$$\therefore p(-1) = (-1 + 1)(-1 - 2)$$
$$\Rightarrow p(-1) = 0 × -3 = 0$$

$$\therefore x = -1$$ is zero of given polynomial.

To check, if x = 2 is zero of given polynomial then p(2) should be equal to 0.

$$\therefore p(2) = (2 + 1)(2 - 2)$$
$$\Rightarrow p(2) = 3 × 0 = 0$$

Therefore, x = 2 is too, a zero of given polynomial.

v) $$p(x) = x^2$$

To check, if x = 0 is zero of given polynomial then p(0) should be equal to 0.

$$\therefore p(0) = (0^2) = 0$$

Therefore, x = 0 is zero of given polynomial.

vi) $$p(x) = lx + m$$

To check, if $$x = \frac{-m}{l}$$ is zero of given polynomial then $$p(\frac{-m}{l} )$$ should be equal to 0.

$$\therefore p(\frac{-m}{l} ) = l(\frac{-m}{l} ) + m$$
$$\Rightarrow p(\frac{-m}{l} )= -m + m = 0$$
$$\therefore, x = \frac{-m}{l}$$ is zero of given polynomial.

vii)$$p(x) = 3x^2 - 1$$

To check, if$$x = \frac{-1}{\sqrt{3}}$$ is zero of given polynomial then p($$\frac{-1}{\sqrt{3}}$$) should be equal to 0.

$$\therefore p(\frac{-1}{\sqrt{3}} ) = 3(\frac{-1}{\sqrt{3}} )^2 - 1$$
$$\Rightarrow p(\frac{-1}{\sqrt{3}} ) = 1 - 1 = 0$$

$$\therefore x = \frac{-1}{\sqrt{3}}$$ is zero of given polynomial.

To check, if $$x = \frac{2}{\sqrt{3}}$$ is zero of given polynomial then $$p(\frac{2}{\sqrt{3}}$$) should be equal to 0.

$$\therefore p(\frac{2}{\sqrt{3}} ) = 3(\frac{2}{\sqrt{3}} )^2 - 1$$
$$\Rightarrow p(\frac{2}{\sqrt{3}} ) = 3$$

$$\therefore, x = \frac{2}{\sqrt{3}}$$ is not a zero of given polynomial.

viii)$$p(x) = 2x + 1$$

To check, if $$x = \frac{1}{2}$$ is zero of given polynomial then$$p(\frac{1}{2} )$$ should be equal to 0.

$$\therefore p(\frac{1}{2} ) = 2(\frac{1}{2} ) + 1$$
$$\Rightarrow p(\frac{1}{2} ) = 1 + 1 = 2$$

$$\therefore x = \frac{1}{2}$$ is not a zero of given polynomial.