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Q.1. Find the remainder when x^{3}+3x^{2}+3x+1 is divided by
(i) x+1
Solution:
x+1= 0
⇒ x = −1
∴ Remainder:
p(−1) = (−1)^{3}+3(−1)2+3(−1)+1
= −1+3−3+1
= 0
(ii) x−1/2
Solution:
x1/2 = 0
⇒ x = 1/2
∴ Remainder:
p(1/2) = (1/2)^{3}+3(1/2)^{2}+3(1/2)+1
= (1/8)+(3/4)+(3/2)+1
= 27/8
(iii) x
Solution:
x = 0
∴ Remainder:
p(0) = (0)^{3}+3(0)^{2}+3(0)+1
= 1
(iv) x+π
Solution:
x+π = 0
⇒ x = −π
∴ Remainder:
p(0) = (−π)^{3} +3(−π)^{2}+3(−π)+1
= −π^{3}+3π^{2}−3π+1
(v) 5+2x
Solution:
5+2x=0
⇒ 2x = −5
⇒ x = 5/2
∴ Remainder:
(5/2)^{3}+3(5/2)^{2}+3(5/2)+1 = (125/8)+(75/4)(15/2)+1
= 27/8
Q.2. Find the remainder when x^{3}−ax^{2}+6x−a is divided by xa.
Solution:
Let p(x) = x^{3}−ax^{2}+6x−a
x−a = 0
∴ x = a
Remainder:
p(a) = (a)^{3}−a(a^{2})+6(a)−a = a^{3}−a^{3}+6a−a = 5a
Q.3. Check whether 7+3x is a factor of 3x3+7x.
Solution:
7+3x = 0
⇒ 3x = −7
⇒ x = 7/3
∴ Remainder:
3(7/3)^{3}+7(7/3) = (343/9)+(49/3)
= (343(49)3)/9
= (343147)/9
= 490/9 ≠ 0
∴ 7+3x is not a factor of 3x3+7x
Check out the NCERT Solutions of all the exercises of Polynomials:
Exercise 2.1. NCERT Solutions: Polynomials
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