Class 9 - Mathematics
Polynomials - Exercise 2.3
Top Block 1
Exercise 2.3
Question : Find the remainder when x3 + 3x2 + 3x + 1 is divided by
(i) x + 1 (ii) x – 1/2 (iii) x (iv) x + π (v) 5 + 2x
Answer :
Let p(x) = x3 + 3x2 + 3x + 1
(i) put x + 1 = 0, we get
x = -1
Using remainder theorem, when p(x) = x3 + 3x2 + 3x + 1 is divided by x + 1, remainder is given
by p(-1).
Now, p(-1) = (-1)3 + 3 * (-1)2 + 3 * (-1) + 1
= -1 + 3 – 3 + 1
= -4 + 4
= 0
Hence, the remainder is 0.
(ii) put x – 1/2 = 0, we get
x = 1/2
Using remainder theorem, when p(x) = x3 + 3x2 + 3x + 1 is divided by x – 1/2, remainder is given
by p(1/2).
Now, p(1/2) = (1/2)3 + 3 * (1/2)2 + 3 * (1/2) + 1
= 1/8 + 3 * 1/4 + 3/2 + 1
= 1/8 + 3/4 + 3/2 + 1
= (1 * 1 + 3 * 2 + 3 * 4 + 1 * 8)/8 [LCM(8, 4, 2, 1) = 8]
= (1 + 6 + 12 + 8)/8
= 27/8
Hence, the remainder is 27/8
(iii) put x = 0
Using remainder theorem, when p(x) = x3 + 3x2 + 3x + 1 is divided by x, remainder is given
by p(0).
Now, p(-1) = (0)3 + 3 * (0)2 + 3 * (0) + 1
= 0 + 0 + 0 + 1
= 1
Hence, the remainder is 1.
(iv) put x + π = 0
Using remainder theorem, when p(x) = x3 + 3x2 + 3x + 1 is divided by x + π, remainder is given
by p(-π).
Now, p(-π) = (-π)3 + 3 * (-π)2 + 3 * (-π) + 1
= -π3 + 3 π3 – 3 π + 1
Hence, the remainder is -π3 + 3 π3 – 3 π + 1.
(v) put 5 + 2x = 0, we get
2x = -5
⇒ x = -5/2
Using remainder theorem, when p(x) = x3 + 3x2 + 3x + 1 is divided by 5 + 2x, remainder is given
by p(-5/2).
Now, p(-5/2) = (-5/2)3 + 3 * (-5/2)2 + 3 * (-5/2) + 1
= -125/8 + 3 * 25/4 – 15/2 + 1
= -125/8 + 75/4 – 15/2 + 1
= (-1 * 125 + 75 * 2 – 15 * 4 + 1 * 8)/8 [LCM(8, 4, 2, 1) = 8]
= (-125 + 150 – 60 + 8)/8
= -27/8
Hence, the remainder is 27/8
Mddle block 1
Question : 2: Find the remainder when x3 – ax2 + 6x – a is divided by x – a.
Answer :
Let p(x) = x3 – ax2 + 6x – a
Put x – a = 0, we get
x = a
Using remainder theorem, when p(x) = x3 – ax2 + 6x – a is divided by x – a, remainder is given
by p(a).
Now, p(a) = a3 – a * a2 + 6 * a – a
= a3 – a3 + 6a – a
= 5a
Hence, the remainder is 5a.
Question : 3: Check whether 7 + 3x is a factor of 3x2 + 7x
Answer :
Given, 3x2 + 7x = 3 * x * x + 7 * x
= x(3 * x + 7)
= x(3x + 7)
= x(7 + 3x)
Hence, 7 + 3x is a factor of 3x2 + 7x