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:

x-1/2 = 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

= −π3+3π2−3π+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