Question

can anyone help me to solve this ​

Answer 1

Step-by-step explanation:

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

1st solution:- The divisor here is x + 1 = x-(-1)

So, by remainder theorem, when p(x) is divided by x+1, the remainder is p(-1)

➠p(x) = x³ + 3x² + 3x + 1

➠ p(-1) = (-1)³ + 3(-1)² + 3(-1) + 1

= -1 + 3 - 3 + 1

= 0

Hence, the required remainder is 0.

2nd Solution:- The divisor here is x-½ [From x - a with a = ½

So, by remainder theorem, when p(x) is divided by x-½, the remainder is p(½).

➠ p(x) = x³ + 3x² + 3x + 1

➠ p(½) = (½)³+ 3(½) + 3(½) + 1

= ⅛ + ¾ + ³/² + 1

= 27/8

Hence, the required remainder is 27/8.

3rd Solution:- The divisor here is x = x - 0

So, by remainder theorem, when p(x) is divided by x, the remainder is p(0).

➠ p(x) = x³ + 3x² + 3x + 1

➠ p(0) = (0)³ + 3(0)² + 3(0) + 1

= 1

Hence, the required remainder is 1.

4th Solution:- The divisor here is x + π = x-(-π). [From x - a with a = -π]

So, by remainder theorem, when p(x) is divided by x+π, the remainder is p(-π).

➠ p(x) = x³ + 3x² + 3x + 1

➠ p(-π) = (-π)³ + 3(-π) + 3(-π) + 1

= -π³ + 3π² - 3π + 1

Hence, the required remainder is -π³ + 3π² - 3π + 1

5th Solution:- The divisor here is 5+2x = 2(x +5/2) = 2{x-(-5/2)}

So, by remainder theorem, when p(x) is divided by 2{x-(-5/2)} the remainder is p(-5/2).

➠ p(x) = x³ + 3x² + 3x + 1

➠ p(-5/2) = (-5/2)³ + 3(-5/2)² + 3(-5/2) + 1

= -125/8 + 3 × 25/4 - 15/2 + 1

= -27/8

Hence the required remainder of -27/8.

:)