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Verify whether the following are True or False :

i)-3 is a zero of x-3

ii) $-\frac{1}{3}$ is a zero of $3x+1$

iii) $-\frac{4}{5}$ is a zero of $4-5y$

iv) 0 and 2 are the zeroes of $t^{2}-2t$

v) –3 is a zero of $y^{2}+y-6$

Answers (1)

i) False

Solution:- Zeroes of a polynomial can be defined as points where the value of the polynomial becomes zero as a whole.

Let,

p(x) = x-3

Putting x = -3

p(x) = $-3-3=-6\neq 0$

Hence -3 is not a zero of x - 3

Therefore given statement is False.

ii) True

Solution :- Zeroes of a polynomial can be defined as points where the value of the polynomial becomes zero as a whole.

Let,

p(x) = 3x+1

Putting x = $-\frac{1}{3}$

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

Hence $-\frac{1}{3}$ is a zero of 3x+1.

Therefore given statement is True.

iii) False

Solution :- Zeroes of a polynomial can be defined as points where the value of the polynomial becomes zero as a whole.

Let,

p(y) = $4-5y$

Putting y = $-\frac{4}{5}$

$p(y) =4-5\left ( \frac{-4}{5} \right )=4+4=16\neq 0$

Hence $- \frac{4}{5}$ is not a zero of 4-5y

Therefore given statement is False.

iv) True

Solution

Method 1: Zeroes of a polynomial can be defined as points where the value of the polynomial becomes zero as a whole.

Let,

$p(t) =t^{2}-2t$

Putting t = 0

$p(t) =0^{2}-2(0)=0$

Putting t = 2

$p(t) =2^{2}-2(2)=0$

Hence 0 and 2 are the zeroes of $t^{2}-2t$

Therefore given statement is True.

Method 2:

$t^{2}-2t=0$

The given expression can be factorized as:

$t(t-2)=0$

Hence, t = 0, t – 2 = 0

So we get the zeroes as t=0,t=2

Therefore given statement is True.

v) True

Solution

Method 1: Zeroes of a polynomial can be defined as points where the value of the polynomial becomes zero as a whole.

Let,

$P(y)=y^{2}+y-6$

Putting y = -3

$P(y)=(-3)^{2}+(-3)-6=9-3-6=0$

Hence -3 is the zero of $y^{2}+y-6$

Therefore given statement is True.

Method 2:

$y^{2}+y-6=0$

The given expression can be factorized as follows:

$\\y^{2}+3y-2y-6=0\\ y(y+3)-2(y+3)=0\\ (y+3)(y-2)=0\\ y=-3,y=2$

Hence zeroes are -3 and 2

Therefore the given statement is True.

Posted by

infoexpert24

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