To find zeroes of a polynomial, we have to equate the polynomial to zero and solve for the variable.
To check whether 'k' is a zero of the polynomial f(x), we have to substitute the value 'k' for 'x' in f(x). If f(k) = 0, then 'k' is a zero of the polynomial f(x).
Question 1 :
Verify whether the following are zeros of the polynomial indicated against them, or not.
(i) p(x) = 2x - 1, x = 1/2
Solution :
p(1/2) = 2(1/2) - 1
= 1 - 1
p(1/2) = 0
Since we get 0 by applying 1/2, we may decide that 1/2 is the zero of the polynomial.
(ii) p(x) = x^{3} - 1, x = 1
Solution :
p(x) = x^{3} - 1
p(1) = 1^{3} - 1
p(1) = 0
Since we get 0 by applying 1, we may decide that 1 is the zero of the polynomial.
(iii) p(x) = ax + b, x = -b/a
Solution :
p(x) = ax + b
p(-b/a) = a(-b/a) + b
p(-b/a) = 0
Since we get 0 by applying -b/a, we may decide that -b/a is the zero of the polynomial.
(iv) p(x) = (x + 3) (x - 4), x = 4, x = –3
Solution :
p(x) = (x + 3) (x - 4)
x = 4 p(4) = (4 + 3) (4 - 4) p(4) = 0 |
x = -3 p(-3) = (-3 + 3) (-3 - 4) p(-3) = 0 |
Hence 4 and -3 are the zeroes of the polynomial.
Question 2 :
Find the number of zeros of the following polynomials represented by their graphs.
(i)
The graph intersects the x axis at two points. Hence number of zeroes is 2.
(ii)
The graph intersects x-axis at 3 points, hence the number of zeroes is 3.
(iii)
The graph does not intersect x axis. So the polynomial will not have solution.
The graph intersects the x-axis at one point. Hence the polynomial will have 1 zero.
(v)
The graph intersects the x-axis at one point. Hence the polynomial will have 1 zero.
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