When the polynomial x3+ 3x2+ 3x + 1 is divided by x + 1, the remainder is :-a)1b)8c)0d)- 6Correct answer is option 'C'. Can you explain this answer?

Question

Answer

The zero of x + 1 is –1

And by remainder theorem, when

p(x) = x³ + 3x² + 3x + 1 is divided by x + 1, then remainder is p(–1).

∴ p(–1) = (–1)³ + 3 (–1)² + 3(–1) + 1

= –1 + (3 × 1) + (–3) + 1

= –1 + 3 – 3 + 1

= 0

Thus, the required = 0

Answer

G(X) = 0
(X+1)= o
X= -1
Now ,
P (X) = X^3+ 3x^2 + 3x +1
p ( -1) = (-1)'3 +3(-1)'2 +3(-1)+1
p(-1) = -1. +3. -3. +1
p(-1). = 0

Answer

Option - c

Answer

By using remainder theorem
x+1=0
x=-1
p(x)= x³+3x²+3x+1
p(-1)=(-1)³+3(-1)²+3(-1)+1
=-1+3-3+1
=0

Answer

When a polynomial is divided by another polynomial, the remainder is found by using the remainder theorem. According to the remainder theorem, if a polynomial f(x) is divided by x-a, the remainder is equal to f(a).

In this case, we are dividing the polynomial x^3 + 3x^2 + 3x + 1 by x + 1. To find the remainder, we substitute -1 for x in the polynomial:

f(-1) = (-1)^3 + 3(-1)^2 + 3(-1) + 1
= -1 + 3 + (-3) + 1
= 0

So, the remainder is 0.

Let's go through the steps in detail:

1. Polynomial Division:
- Divide the first term of the dividend (x^3) by the first term of the divisor (x). This gives us x^2.
- Multiply the divisor (x + 1) by the quotient (x^2). This gives us x^3 + x^2.
- Subtract this product from the dividend (x^3 + 3x^2 + 3x + 1):
(x^3 + 3x^2 + 3x + 1) - (x^3 + x^2) = 2x^2 + 3x + 1.

2. Repeat the process:
- Divide the first term of the new dividend (2x^2) by the first term of the divisor (x). This gives us 2x.
- Multiply the divisor (x + 1) by the new quotient (2x). This gives us 2x^2 + 2x.
- Subtract this product from the new dividend (2x^2 + 3x + 1):
(2x^2 + 3x + 1) - (2x^2 + 2x) = x + 1.

3. Final step:
- Divide the first term of the new dividend (x) by the first term of the divisor (x). This gives us 1.
- Multiply the divisor (x + 1) by the new quotient (1). This gives us x + 1.
- Subtract this product from the new dividend (x + 1):
(x + 1) - (x + 1) = 0.

The remainder is 0, which means that the polynomial x^3 + 3x^2 + 3x + 1 is evenly divisible by x + 1. Therefore, option C is the correct answer.

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