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3.Verify whether the following are zeroes of the polynomial, indicated against them.
i)\(p(x) = 3x + 1\), x = -1/3
ii)\(p(x) = 5x - \pi\), x = 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 = -m/l
vii)\(p(x) = 3x^2 - 1\), \(x = -1/\sqrt{3}\), \(2/\sqrt{3}\)
viii)\(p(x) = 2x + 1\), x = 1/2

Answer :

i)\(p(x) = 3x + 1\)
To check, if x = -1/3 is zero of given polynomial then p(-1/3) should be equal to 0.
So, p(-1/3) = 3(-1/3) + 1 = -1 +1 = 0
Therefore, x = -1/3 is zero of given polynomial.

ii)\(p(x) = 5x - \pi\)
To check, if 4/5 is zero of given polynomial then p(4/5) should be equal to 0.
So, \(p(4/5) = 5(4/5) - \pi = 4 - \pi\)
Therefore, x = 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.
So, p(1) = \(1^2 - 1\) = 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.
So, p(-1) = \({-1}^2 - 1\) = 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.
So, p(-1) = (-1 + 1)(-1 - 2) = 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.
So, p(2) = (2 + 1)(2 - 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.
So,\( p(0) = (0^3) = 0\)
Therefore, x = 0 is zero of given polynomial.

vi)\(p(x) = lx + m\)
To check, if x = -m/l is zero of given polynomial then p(-m/l) should be equal to 0.
So, p(-m/l) = l(-m/l) + m = -m + m = 0
Therefore, x = -m/l is zero of given polynomial.

vii)\(p(x) = 3x^2 - 1\)
To check, if\( x = -1/\sqrt{3}\) is zero of given polynomial then p(\(-1/\sqrt{3}\)) should be equal to 0.
So, p(\(-1/\sqrt{3}\)) = \(3(-1/\sqrt{3})^2 - 1\) = 1 - 1 = 0
Therefore,\( x = -1/\sqrt{3}\) is zero of given polynomial.
To check, if \(x = 2/\sqrt{3}\) is zero of given polynomial then p(\(2/\sqrt{3}\)) should be equal to 0.
So, p(\(2/\sqrt{3}\)) = \(3(2/\sqrt{3})^2 - 1\) = 3
Therefore, \(x = 2/\sqrt{3}\) is not a zero of given polynomial.

viii)\(p(x) = 2x + 1\)
To check, if x = 1/2 is zero of given polynomial then p(1/2) should be equal to 0.
So, p(1/2) = 2(1/2) + 1 = 1 + 1 = 2
Therefore, x = 1/2 is not a zero of given polynomial.