Question

What must be added to \[{{x}^{4}}+2{{x}^{3}}-2{{x}^{2}}+x-1\] so that the result is exactly divisible by \[{{x}^{2}}+2x-3\].

Answer 1

Hint: To solve such questions we first assume both the given polynomials as some functions f(x) or g(x) depending upon if one or two variables are used in the given polynomials then the question can be solved easily.

Complete Step-by-Step solution:
We have to calculate that what must be added to \[{{x}^{4}}+2{{x}^{3}}-2{{x}^{2}}+x-1\] so that the result is exactly divisible by \[{{x}^{2}}+2x-3\].
First of all, we assume the given two polynomials as f(x) and g(x) as both of them are the functions of x.
Let \[f(x)={{x}^{4}}+2{{x}^{3}}-2{{x}^{2}}+x-1\] be the polynomial function in which some term is to be added and let\[g(x)={{x}^{2}}+2x-3\], the other polynomial.
Now we will divide the polynomial f(x) by g(x) so as to obtain a remainder.
Dividing f(x) by g(x) we have,
\[\dfrac{{{x}^{4}}+2{{x}^{3}}-2{{x}^{2}}+x-1}{{{x}^{2}}+2x-3}\]
The above term on dividing gives the quotient as \[{{x}^{2}}+1\] and the remainder as \[-x+2\].
Hence, we obtain the remainder when f(x) is divided by g(x) as -x+2.
Let us assume a new function r(x) which becomes equal to the obtained remainder.
Hence, \[r(x)=-x+2\]
Therefore, -r(x) must be added to the f(x) to make f(x) exactly divisible by g(x).
We have used a negative sign before r(x) because we had to add the number or the polynomial to f(x) and not subtract.
We have -r(x) is,
\[\begin{align}
  & -r(x)=-[-x+2] \\
 & \Rightarrow -r(x)=x-2 \\
\end{align}\]
 Therefore, x - 2 should be added to \[{{x}^{4}}+2{{x}^{3}}-2{{x}^{2}}+x-1\] so that the result is exactly divisible by\[{{x}^{2}}+2x-3\].

Note: The possibility of error in this question can be at the point where the remainder is obtained and you must be wondering about whether to apply a negative sign in front of it or not. We always apply a negative sign when we know that the obtained remainder would be added to the polynomial so as to be completely divisible by the given polynomial.