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, -1
iv)\(p(x) = (x + 1)(x - 2)\), x = -1, 2
v)\(p(x) = x^2\), x = 0
vi)\(p(x) = lx + m, x = \frac{-m}{l} \)
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.