Polynomials

Exercise 2.2 Part 3

Question 3: Verify whether the following are zeros of the polynomial, indicated against them.

(i) `p(x)=3x+1, x=-1/3`

Solution: Given, `p(x)=3x+1`

When `x=-1/3`

Then `p(-1/3)=3xx(-1/3)+1`

Or, `p(-1/3)=-1+1`

Or, `p(-1/3)=0`

Answer: Yes


(ii) `p(x)=5x-π, x=4/5`

Solution: Given `p(x)=5x-π`

When `x=4/5`

Then `p(4/5)=5xx4/5\-π`

Or, `p(4/5)=4-π`

Or, `p(4/5)≠0`

Answer: No

(iii) `p(x) = x^2 - 1, x =1, - 1`

Answer: Given, `p(x) = x^2 - 1`

At p(1), i.e. `x =1`

`p(1) = 1^2 -1`

Or, `p(1) = 1 – 1 = 0`

At p( - 1), i.e. `x = - 1`

`p( -1) = ( -1)^2 -1`

Or, `p( -1) = 1 – 1 = 0`

Therefore, both 1 and -1 are zeroes of the given polynomial.

(iv) `p(x) = (x + 1)(x – 2), x = -1, 2`

Answer: When `x = - 1`

Then, `p(1) = ( - 1 + 1)( - 1 – 2)`

Or, `p(1) = 0 x ( - 3) = 0`

When `x = 2`

Then `p(2) = (2 + 1)(2 – 2)`

Or, `p(2) = 3 xx 0 = 0`

Theefore, both – 1 and 2 are zeroes of the given polynomial.


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

Answer: Given, `p(x) = x^2`

When `x = 0`

Then `p(0) = 0^2`

Or, `p(0) = 0`

Therefore, 0 is the zero of the given polynomial.

(vi) `p(x)=lx+m` where `x=-m/l`

Solution: When `x=-m/l`

Then `p(-m/l)=-(m)/(l)xxl+m`

Or, `p(-m/l)=-m+m`

Or, `p(-m/l)=0`

Answer: Yes

(vii) `p(x)=3x^3-1` where `x=-1/sqrt3\,2/sqrt3`

Solution: When `x=-1/sqrt3`

Then `p(-1/sqrt3)=3(-1/sqrt3)^2-1`

Or, `p(-1/sqrt3)=3xx1/3\-1`

Or, `p(-1/sqrt3)=1-1=0`

When `x=2/sqrt3`

Then `p(2/sqrt3)=3(2/sqrt3)^2-1`

Or, `p(2/sqrt3)=3xx4/3\-1`

Pr, `p(2/sqrt3)=4-1=3`

Thus `-1/sqrt3` is the zero of polynomial.

(viii) `p(x)=2x+1` where `x=1/2`

Solution: When `x=1/2`

Then `p(1/2)=2xx1/2\+1`

Or, `p(1/2)=1+1=2`

Thus, ½ is not the zero of the polynomial.



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