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Polynomials

    Exercise 2.2

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9th class > Polynomials > Exercise 2.2

Exercise 2.2

  1. Find the value of the polynomial 5x−4x2+3 at:

Instructions

(i)x = 0
Substituting 0 in the equation: = 50−402+3
= - + 3 = .
(ii) x = –1
Substituting −1 in the equation: 5−1−4−12+3 = + + 3
= -5 -4 +3 =
(iii) x = 2
Substituting 2 in the equation: 52−422+3 = - + 3
= 10 -16 +3 =
  1. Find p(0), p(1) and p(2) for each of the following polynomials.

Instructions

(i) p(y) = y2−y+1
Substituting 0, 1 and 2 in equation p(y).
p(0) = 02−0+1 = .
p(1) = 12−1+1 = .
p(2) = 22−2+1 = .
(ii) p(t) = 2+t+2t2−t3
Substituting 0, 1 and 2 in equation p(t).
p(0) = 2+0+202−03 = .
p(1) = 2+1+212−13 = .
p(2) = 2+2+222−23 =.
(iii) p(x) = x3
Substituting 0, 1 and 2 in equation p(x).
p(0) = 03 = .
p(1) = 13 = .
p(2) = 23 = .
(iv) p(x) = x−1x+1
Substituting 0, 1 and 2 in equation p(x).
We can write p(x) = x2−1 using identity property a+ba−b = a2−b2
p(0) = 02−1 =
p(1) = 12−1 =
p(2) = 22−1 =

3. Verify whether the following are zeroes of the polynomial, indicated against them.

Instructions

(i) p(x) = 3x+1, x = −13
p−13 = 3 × () + 1 = .
Therefore, −13 is a zero of p(x).
(ii) p(x) = 5x−π , x=45
p45 = 5() - π with π = 227 which gives us p45 = 4−227 = .
Therefore, 45 is not a zero of p(x).
(iii) p(x) = x2−1, x = -1, 1
p(-1) = ^2 - 1 = .
p(1) = ^2 - 1 = .
Therefore, 1 and -1 are zeroes of p(x).
(iv) p(x) = x+1x−2, x = -1, 2
For p−1 = ( + 1)( - 2) = × = .
For p2 = ( + 1)( - 2) = × = .
Therefore, -1 and 2 are zeroes of p(x).
(v) p(x) = x2, x = 0
p(0) = = .
Therefore, 0 is a zero of p(x).
(vi) p(x) = lx+m, x =−ml
p−ml = l × () + m = + = 0.
Therefore, −ml is a zero of p(x).
(vii) p(x) = 3x2−1, x = −13, 23
For p(−13) = 3×132−1 = 3 × () - 1 = .
Therefore,−13 is a zero of p(x).
For p(23) = 3×232−1 = 343−1 = .
Therefore, 23 is not a zero of p(x).
(viii) p(x) = 2x+1, x = 12
p12 = 2() + 1 =
Therefore, 12 is not a zero of p(x).
  1. Find the zero of the polynomial in each of the following cases.

(i) p(x) = x + 5 = 0 : x = is the zero.

(ii) p(x) = x – 5 = 0 : x = is the zero.

(iii) p(x) = 2x + 5 = 0 : x = is the zero.

(iv) p(x) = 3x – 2 = 0 : x = is the zero.

(v) p(x) = 3x = 0 : x = is the zero.

(vi) p(x) = ax , a ≠ 0: x = is the zero.

(vii) p(x) = cx + d, c ≠ 0, c, d are real numbers, cx + d = 0 : x = is the zero.