Question

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

Answer 1

Hint: We know from the basic division formula that the quotient when multiplied and added to remainder the result would be the same as the divisor. The question asked here is indirectly asked the remainder which obtain when \[{{x}^{4}}+2{{x}^{3}}-2{{x}^{2}}+x-1\] is divided by \[{{x}^{2}}+2x-3\]. To do so describe the given equations as functions of \[x\] .

Complete step-by-step solution:
Let us consider the equations as the functions of \[x\] given by
\[f(x)={{x}^{4}}+2{{x}^{3}}-2{{x}^{2}}+x-1 \]
 \[g(x)={{x}^{2}}+2x-3 \]
Consider \[r(x)\] as the remainder
We need to divide \[f(x)\] with \[g(x)\]
It can be done as below
\[r(x)=\dfrac{f(x)}{g(x)}\]
The higher power of \[f(x)\] is four and of \[g(x)\] is two.
So, multiply the equation \[g(x)\] with \[{{x}^{2}}\] we get
\[\Rightarrow {{x}^{2}}\left( {{x}^{2}}+2x-3 \right)={{x}^{4}}+2{{x}^{3}}-3{{x}^{2}}\,\,\,\,\,\,\,\,\,\]
Subtract this product from \[f(x)\]
\[ f(x)-\left( {{x}^{4}}+2{{x}^{3}}-3{{x}^{2}} \right)={{x}^{4}}+2{{x}^{3}}-2{{x}^{2}}+x-1-\left( {{x}^{4}}+2{{x}^{3}}-3{{x}^{2}} \right)\]
\[ \Rightarrow f(x)-\left( {{x}^{4}}+2{{x}^{3}}-3{{x}^{2}} \right)={{x}^{4}}+2{{x}^{3}}-2{{x}^{2}}+x-1-{{x}^{4}}-2{{x}^{3}}+3{{x}^{2}} \]
\[ \Rightarrow f(x)-\left( {{x}^{4}}+2{{x}^{3}}-3{{x}^{2}} \right)={{x}^{2}}+x-1 \]
The remainder is \[{{x}^{2}}+x-1\] .
This equation can also be divided by \[g(x)\] as the power of the obtained equation is equal to the power of \[g(x)\] .
So, we consider \[g(x)\times 1\] subtracted from obtained equation
\[\left( {{x}^{2}}+x-1 \right)-g(x)=\left( {{x}^{2}}+x-1 \right)-\left( {{x}^{2}}+2x-3 \right) \]
\[ \Rightarrow \left( {{x}^{2}}+x-1 \right)-g(x)={{x}^{2}}+x-1-{{x}^{2}}-2x+3 \]
\[ \Rightarrow \left( {{x}^{2}}+x-1 \right)-g(x)=-x+2 \]
Hence the required equation \[r(x)=-x+2\]
Hence it is this equation we need to add to \[f(x)\] so as to be exactly divided by the expression \[g(x)\]
Additional Information: The division between polynomials can be done with the help of a long division method which is similar to normal division. the divisor should be multiplied to the highest degree so that it is equal to the degree of the polynomial.

Note: The division of polynomials is different from the normal division of real numbers in the aspect of the multiple elements. In polynomial division the quotient must be multiplied with each element of the expression of divisor. If we add \[r(x)\] to \[f(x)\] and divide the sum with \[g(x)\] . then it will divide exactly.