Question

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 – π, 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 = -$\frac{1 }{ √3} ,\frac{2}{ √3}$

(viii) p(x) = 2x + 1, x = $\frac{1}{2}$

Answer 1

(i) If x = -$\frac{1}{3}$ is a zero of given polynomial p(x) = 3x + 1, then p(-$\frac{1}{3}$) should be 0.

Here, p (-$\frac{1}{3}$) = 3 (-$\frac{1}{3}$) + 1 = -1 + 1 = 0.

Therefore, x = -$\frac{1}{3}$ is a zero of the given polynomial.

(ii) If x = $\frac{4}{5}$ is a zero of polynomial p(x) = 5x - π, then p(4/5) should be 0.

Here, p($\frac{4}{5}$) = 5 ($\frac{4}{5}$) - π = 4 - π As p (4/5) ≠ 0,

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

(iii) If x = 1 and y = -1 are zeroes of polynomial p(x) = x2 - 1, then p(1) and p(-1) should be 0.

Here, p(1) = (1)2 - 1 = 0, and p(-1) = (-1)2 - 1 = 0

Hence, x = 1 and −1 are zeroes of the given polynomial.

(iv)If x = −1 and x = 2 are zeroes of polynomial p(x) = (x +1) (x − 2), then p(−1) and p(2)should be 0.

Here, p(−1) = (− 1 + 1) (− 1 − 2) = 0 (−3) = 0, and p(2) = (2 + 1) (2 − 2 ) = 3 (0) = 0

Therefore, x = −1 and x = 2 are zeroes of the given polynomial.

(v) If x = 0 is a zero of polynomial p(x) = x2 , then p(0) should be zero.

Here, p(0) = (0)2 = 0.

Hence, x = 0 is a zero of the given polynomial.

(vi) If x = -$\frac{m }{ l}$is a zero of polynomial p(x) = lx + m, then should be 0.

Here, p(-$\frac{m }{ l}$ )= (-$\frac{m }{ l}$) + m = -m + m = 0.

Therefore, x = -$\frac{m }{ l}$l is a zero of the given polynomial.

(vii) If x = -$\frac{1 }{ √3}$and x = $\frac{2}{ √3}$ are zeroes of polynomial

p(x) = 3x2 -1 , then p(-$\frac{m}{l}$)

p(-$\frac{1}{√3}$) and p ($\frac{2}{√3}$) should be 0.

Here, p(-$\frac{1}{√3}$) = 3 (-$\frac{1}{√3}$)2 -1 = 3 ($\frac{1}{3}$)-1 = 1-1 = 0, and

p(2/√3) = 3 ($\frac{2}{√3}$)3 - 1 = 3 ($\frac{2}{√3}$)2 -1 = 3($\frac{4}{3}$) - 1 = 4 - 1 = 3.

Hence, x = -$\frac{1}{√3}$ is a zero of the given polynomial.

However, x = 2√3 is not a zero of the given polynomial.

(viii) If x = $\frac{1}{2}$ is a zero of polynomial p(x) = 2x + 1, then p(1/2) should be 0.

Here, p ($\frac{1}{2}$) = 2($\frac{1}{2}$) + 1 = 1 +1 = 2. As p ≠ 0,

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