Polynomials (Exercise 2.3) EX 2.3 QUESTION 1.

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
Solution:

(i) x+1= 0

⇒x = −1

p(−1) = (−1)3+3(−1)2+3(−1)+1

= −1+3−3+1

= 0

∴Remainder = 0

(ii) x- 1/2

x-1/2 = 0

⇒ x = 1/2

p(1/2) = (1/2)3+3(1/2)2+3(1/2)+1

= (1/8)+(3/4)+(3/2)+1

= 27/8

∴Remainder = 27/8

(iii) x = 0

p(0) = (0)3+3(0)2+3(0)+1

= 1

∴Remainder = 0

(iv)

x+π = 0

⇒ x = −π

p(0) = (−π)+3(−π)2+3(−π)+1

= −π3+3π2−3π+1

∴Remainder = −π3+3π2−3π+1

5+2x=0

⇒ 2x = −5

⇒ x = -5/2

(-5/2)3+3(-5/2)2+3(-5/2)+1 = (-125/8)+(75/4)-(15/2)+1

= -27/8

∴Remainder =-27/8

EX 2.3 QUESTION 2.

Find the remainder when x3−ax2+6x−a is divided by x-a.

Solution:

Let p(x) = x3−ax2+6x−a

x−a = 0

∴x = a

p(a) = (a)3−a(a2)+6(a)−a

= a3−a3+6a−a = 5a

∴ Remainder = a3−a3+6a−a = 5a

EX 2.3 QUESTION 3.

Check whether 7+3x is a factor of 3x3+7x.

Solution:

7+3x = 0

⇒ 3x = −7

⇒ x = -7/3

3(-7/3)3+7(-7/3)

= -(343/9)+(-49/3)

= (-343-(49)3)/9

= (-343-147)/9

= -490/9 ≠ 0

∴Remainder ≠ 0

∴7+3x is not a factor of 3x3+7x