Question

using Ram and remainder theorem find the remainder when x^3+x^2+x+1 us divided by x+1​

Answer 1

Correct Question -

Using remainder theorem find the remainder when x^3+x^2+x+1 is divided by x+1

Given -

• $\sf p(x) \: = {x}^{3} \: + \: {x}^{2} \: + \: x \: + \: 1$
• $\sf \: g(x) = x \: + \: 1$

To find -

• Remainder using remainder theorem.

Solution -

In the question, we are provided with a polynomial expression, and we need to divid it with x + 1, for that first we will find the zero of g(x), after that we will put the obtained value in place of x, that will give us the remainder of polynomial p(x). Let's do it!

So -

Let's find the zero of x + 1.

$\sf \longrightarrow \: \: x \: + \: 1 \: = 0 \\ \\ \sf \: x \: = 0 \: - \: 1 - - - - (by \: transposition) \\ \\ \sf \longrightarrow \: \: x \: = \: - 1$

Now -

We will find the remaining, by placing -1 in place of x, in the given polynomial.

$\sf \longrightarrow \: p( - 1) \: = { (- 1)}^{3} \: + {( - 1)}^{2} \: + ( - 1) \: + 1 \\ \\ \sf \longrightarrow \: p (- 1) \: = \: - 1 \: + 1 \: - 1 \: + 1 \\ \\ \sf \longrightarrow \: p( - 1) \: = 0 \: ans$

Points to remember -

• While we transpose any number to the other side, it's actually sign changes into it's opposite sign.

• While solving this question, two numbers having opposite signs gets cancelled out, like we have dine in the above question.

• Any constant(number) having negative sign, having, odd number as a power (for eg; 9³,4⁵,7⁹, so on .... ) will have negative sign, and constants having negative sign have the even number as a power (for eg; 3², 5⁶, 8⁸, and so ... ) their negative sign will be changed into positive sign.

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Answer 2

Correct Question -

• Using remainder theorem, find the remainder when x^3+x^2+x+1 is divided by x+1

Given -

$\tt p(x) \: = {x}^{3} \: + \: {x}^{2} \: + \: x \: + \: 1$

$\tt \: g(x) = x \: + \: 1$

To find -

• Remainder using remainder theorem.

Solution :-

Step : 1 To find zero of g(x)

Put g(x) = 0

$: \implies \: \tt \: x \: + \: 1 \: = \: 0$

$: \implies \: \tt \: x \: = \: - \: 1$

Step : 2 To find the remainder when p(x) is divided by g(x)

Using Remainder Theorem, when p(x) is divided by x + 1, the remainder is p(-1).

$: \implies \: \tt \: p( - 1) = {( - 1)}^{ 3} + {( - 1)}^{2} + ( - 1) + 1$

$: \implies \: \tt \: p( - 1) \: - 1 \: + \: 1 \: - 1 \: + \: 1$

$: \implies \: \tt \: p( - 1) \: = \: 0$

$\tt \:  \large \boxed{ \pink{ \tt \: So \: remainder \: is \: 0}}$