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Remainder when \[p\left( x \right) = {x^4} - 5x + 6\] is divided by \[g\left( x \right) = 2 - {x^2}\] is \[ - mx + 2m\]. Find \[m\].

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Last updated date: 20th Jun 2024
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Answer
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Hint:
Here, we have to find the value of \[m\]. First, we will divide the given polynomials to find the quotient and the remainder. Then we will compare the remainder obtained with the given remainder to find the value of \[m\]. Remainder is the amount left after dividing two integers.

Complete step by step solution:
We are given with a polynomial \[p\left( x \right) = {x^4} - 5x + 6\] and a polynomial \[g\left( x \right) = 2 - {x^2}\].
Now we will divide \[p\left( x \right)\] by \[g\left( x \right)\].
Thus, we will get
\[\begin{array}{*{20}{l}}{}&{ - {x^2} - 2}\\{ - {x^2} + 2}&{\left| \!{\overline {\,
 \begin{array}{l}{x^4} + 0{x^3} + 0{x^2} - 5x + 6\\ - \left( {{x^4} + 0{x^3} - 2{x^2}} \right)\end{array} \,}} \right. }\\{}&{\left| \!{\overline {\,
 \begin{array}{l}0 + 2{x^2} - 5x + 6\\\underline { - \left( {2{x^2} + 0x - 4} \right)} \\ - 5x + 10\end{array} \,}} \right. }\end{array}\]
Thus, when dividing \[p\left( x \right)\] by \[g\left( x \right)\], we get the quotient as \[ - {x^2} + 2\] and the remainder as \[ - 5x + 10\].
Now, equating the remainder obtained with the given remainder, we get
\[ - mx + 2m = - 5x + 10\]
Rewriting the above equation, we get
\[ \Rightarrow - mx + 2m = - 5x + 2\left( 5 \right)\]
By comparing the coefficients of \[x\] and the constant term, we get
\[ \Rightarrow - mx = - 5x\] and \[2m = 10\]
\[ \Rightarrow m = 5\] and \[m = \dfrac{{10}}{2}\]
\[ \Rightarrow m = 5\] and \[m = 5\]

Therefore, the value of \[m\] is 5.

Additional Information:
We should follow these steps while dividing a polynomial by another polynomial.
1) First the dividend and the divisor should be arranged in the descending order.
2) We have that the first term of the dividend should be divided by the first term of the divisor, so the answer obtained would be the quotient of the polynomial.
3) Multiplying each term with the first term of the quotient we would get the product of these terms subtracted from the dividend to get the remainder.
4) Now, the first term of the remainder should be divided by the first term of the divisor until the remainder becomes zero or the remainder is the polynomial whose degree is less than the degree of the divisor.

Note:
We should know that when one polynomial is divided by a linear polynomial, then the remainder can be easily found by using the remainder theorem or the factor theorem. To check whether the quotient and the remainder obtained is right, we will use the formula Dividend \[ = \] Quotient \[ \times \] Divisor \[ + \] Remainder.
Thus,
 \[{x^4} - 5x + 6 = \left( { - {x^2} - 2} \right)\left( { - {x^2} + 2} \right) + \left( { - 5x + 10} \right)\]
Multiplying the terms, we get
\[ \Rightarrow {x^4} - 5x + 6 = \left( {{x^4} - 2{x^2} + 2{x^2} - 4} \right) + \left( { - 5x + 10} \right)\]
Adding and subtracting the like terms, we get
\[ \Rightarrow {x^4} - 5x + 6 = \left( {{x^4} - 2{x^2} + 2{x^2} - 4 - 5x + 10} \right)\]
\[ \Rightarrow {x^4} - 5x + 6 = \left( {{x^4} - 5x + 6} \right)\]
Thus the value obtained is right.