# Solve this question? 2x3+x2+2x+1 / x3-x2+x-1Question in test could not answer

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Supposing that you need to reduce the fraction to its lowest terms, you need to factorize both numerator and denominator.

`(2x^3 + x^2 + 2x + 1)/(x^3 - x^2 + x - 1)`

You need to factorize the numerator, hence you need to group the terms such that:

`(2x^3 + x^2) + (2x + 1)= x^2(2x + 1) + (2x + 1) `

You need to factor out 2x + 1 such that:

`2x^3 + x^2 + 2x + 1 = (2x + 1)(x^2 + 1)`

You need to factorize the denominator, hence you need to group the terms such that:

`(x^3 - x^2) + (x + 1) = x^2(x - 1) + (x - 1)`

You need to factor out x - 1 such that:

`x^3 - x^2 + x - 1 = (x - 1)(x^2 + 1)`

You need to write the factored form of fraction such that:

`((2x + 1)(x^2 + 1))/((x - 1)(x^2 + 1))`

Notice that the factored form accentuates the common factors, hence you need to reduce these factors such that:

`((2x + 1)(x^2 + 1))/((x - 1)(x^2 + 1))=(2x + 1)/(x - 1)`

**Hence, simplifying the fraction to its lowest terms yields `(2x^3 + x^2 + 2x + 1)/(x^3 - x^2 + x - 1) = (2x + 1)/(x - 1).` **

2x^3 + x^2 + 2x + 1 / x^3 - x^2 + x + 1

If you assume this problem is actually 2x^3 + x^2 + 2x + 1 / (x^3 - x^2 + x + 1), then the divisor(bottom polynomial) will go into the dividend (top polynomial) 1 time. In order to determine the remainder, you subtract the divisor from the dividend:

2x^3 + x^2 + 2x + 1

- x^3 - x^2 + x + 1

______________________

x^3 + 2X^2 + X

You can then factor out x, which gives you

x(x^2 + 2x + 1)

To find solutions, you must set the equation as equal to zero:

x(x^2 + 2x + 1) = 0

factoring the polynomial, you get

x (x + 1)(X +1) = 0

This gives you three possible answers: x=0; X+1=0; x+1=0

Since the second and third are the same, by solving one: x+1=0, then x= -1

Therefore your possible answers for x are 0 and -1. If you go back to the original equation and replace the variable (x) with either 0 or -1, you will find that either gives a true answer.

Therefore, your answer set is 0, -1

2x^3 + x^2 + 2x + 1 / x^3 - x^2 + x + 1

using long division.... x^3 goes into 2x^3 2 times so you take the dividend and multiply it all by 2 yielding....

2(x^3 - x^2 + x + 1) = 2x^3 - 2x^2 + 2x + 2

You then subtract that from 2x^3 + x^2 + 2x + 1 which yields..

(2x^3 + x^2 + 2x + 1) - (2x^3 - 2x^2 + 2x + 2) = 3x^2 - 1

The 3x^2 - 1 is your remainder so your answer would be either..

2 remainder 3x^2 - 1 or leaving your remainder as a fraction would give you....

2 + ((3x^2 - 1)/(x^3 - x^2 + x + 1))

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