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−$\pi$,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

Answer 1

(i) p(x)=3x+1,x=$\frac{-1}{3}$

p$\left(\frac{-1}{3}\right)$=3$\left(\frac{-1}{3}\right)$+1=−1+1=0

p$\left(\frac{-1}{3}\right)$=0 which means that $\frac{-1}{3}$ is zero of the polynomial p(x)=3x+1.

(ii) p(x)=5x−$\pi$,x=$\frac{4}{5}$

p$\left(\frac{4}{5}\right)$=5$\left(\frac{4}{5}\right)$−$\pi$=4−$\pi$

p$\left(\frac{4}{5}\right)$ $\ne$0 which means that $\frac{4}{5}$ is not zero of the polynomial p(x)=5x−$\pi$.

(iii) p(x)=$x^2$−1,x=1,−1

p(1)=$1^2$−1=1−1=0

p(−1)=${(-1)}^2$−1=1−1=0

Both p(1) and p(−1) are equal to 0. It means that 1 and -1 are zeroes of the polynomial p(x)=$x^2$−1.

(iv) p(x)=(x+1)(x−2),x=−1,2

p(−1)=(−1+1)(−1−2)=0×−3=0

p(2)=(2+1)(2−2)=3×0=0

Both p(−1) and p(2) are equal to 0. It means that -1 and 2 are zeroes of the polynomial p(x)=(x+1)(x−2).

(v) p(x)=$x^2$,x=0

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

p(0)=0 which means that 0 is the zero of the polynomial p(x)=$x^2$.