Question

Find the remainder when
${x}^{3} + 3 {x}^{2} + 3x + 1$
is divided by
$x + \pi$
​

Answer 1

Step-by-step explanation:

Given :-

The polynomial is x³+3x²+3x+1

To find :-

The remainder when x³+3x²+3x+1is divided by x+π

Solution :-

Given cubic polynomial is

P(x) = x³+3x²+3x+1

Given divisor = x+π

We know that

By Remainder Theorem,

If P(x) is divided by x+π then the remainder is P(-π)

Now,

P(-π) = (-π)³+3(-π)²+3(-π)+1

=> P(-π) = -π³+3π²-3π+1

Therefore, The remainder = -π³+3π²-3π+1

Answer :-

The required remainder is -π³+3π²-3π+1

Check:-

x+π ) x³+3x²+3x+1 ( x²+(3-π)x+(3-3π+π²)

x³-πx²

(-) (+)

____________________

(3-π)x²+3x

(3-π)x²+(3π-π²)x

(-) (-)

____________________

(3-3π+π²)x+1

(3-3π+π²)x+(3π-3π²+π³)

(-) (-)

___________________________

-(3π-3π²+π³)+1

____________________________

Remainder = -(3π-3π²+π³)+1

=> Remainder = -3π+3π²-π³+1

Therefore, Remainder = -π³+3π²-3π+1

Verified the given relations in the given problem.

Used Theorem:-

Remainder Theorem:-

" Let P(x) be a polynomial of the degree greater than or equal to 1 and x-a is another linear polynomial, if P(x) is divided by x-a then the remainder is P(a)".

Answer 2

solution :

• please check the attached file