Question

If the polynomial \[{{x}^{4}}-6{{x}^{3}}+16{{x}^{2}}-25x+10\]is divided by another polynomial \[{{x}^{2}}-2x+k\], the remainder comes out to be \[x+a\], find k and a.

Answer 1

Hint: Given that the function is \[{{x}^{4}}-6{{x}^{3}}+16{{x}^{2}}-25x+10\] . We have to divide it with\[{{x}^{2}}-2x+k\] . We have to solve the function with the Division algorithm method which is the same as normal division. It is a step by step procedure to eliminate all the terms and write them in the form of Dividend = Divisor \[\times \] Quotient + Remainder.

Complete step by step answer:
The given function is \[{{x}^{4}}-6{{x}^{3}}+16{{x}^{2}}-25x+10\]. We have to divide it with \[{{x}^{2}}-2x+k\].
First arrange the term of dividend and the divisor in the decreasing order of their degrees.
To obtain the first term of the quotient divide the highest degree term of the dividend by the highest degree term of the divisor.
To obtain the second term of the quotient, divide the highest degree term of the new dividend obtained as remainder by the highest degree term of the divisor.
Continue this process till the degree of remainder is less than the degree of divisor.
\[{{x}^{2}}-2x+k\overset{{{x}^{2}}-4x+8-k}{\overline{\left){{{x}^{4}}-6{{x}^{3}}+16{{x}^{2}}-25x+10}\right.}}\]
                  \[{{x}^{4}}-2{{x}^{3}}+k{{x}^{2}}\]
……………………………………………………………..
                         \[-4{{x}^{3}}+16{{x}^{2}}-k{{x}^{2}}-25x+10\]
                        \[-4{{x}^{3}}+8{{x}^{2}}\] \[-4kx\]
…………………………………………………………..
                                  \[8{{x}^{2}}-k{{x}^{2}}-25x+4kx+10\]
                                         \[8{{x}^{2}}\] \[-16x\] \[+8k\]
. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .
                                       \[-k{{x}^{2}}-9x+4kx+10-8k\]
                                                \[-k{{x}^{2}}\] \[+2kx\] \[-{{k}^{2}}\]
………………………………………………………………………………………………..
                                                        \[-9x+2kx+10-8k+{{k}^{2}}\]
Thus the remainder is \[-9x+2kx+10-8k+{{k}^{2}}\]
\[=\left( 2k-9 \right)x+{{k}^{2}}-8k+{{k}^{2}}\]
In the question it was given that the remainder = \[x+a\]
\[\left( 2k-9 \right)x+{{k}^{2}}-8k+{{k}^{2}}=x+a\]
By comparing,
\[2k-9=1\]
\[k=5\]
\[{{k}^{2}}-8k+10=a\]
\[a={{5}^{2}}-8\left( 5 \right)+10\]
\[a=-5\]
\[k=5\],\[a=-5\]
Note: A proven statement that we use to prove another statement. On the other hand, an algorithm refers to a series of well-defined steps that gives a procedure for solving a type of problem.