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How do you write the remainder in synthetic division?

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Last updated date: 22nd Feb 2024
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IVSAT 2024
Answer
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Hint: We at first check whether the constant term of the linear expression is a root of the polynomial or not. Then, we express the polynomial in terms of the product of its factors and adding the remainder at the end. Upon dividing by the linear expression, the numerator of the fraction remaining is the remainder.

Complete step by step answer:
Synthetic division is a way to divide a polynomial by a linear expression. This is done to find the roots or solution to the equation made by equating the polynomial to zero. First of all, we have to find a value of $x$ which makes the polynomial equal to $0$ . This means $x-\left( that\_value \right)$ is the factor of that polynomial. We then divide the polynomial by that linear expression by synthetic division method to get other roots.
Synthetic division is also used to find whether a polynomial is divisible by a linear expression or not. In general cases, there will be a quotient and a remainder.
Let us take the example of the polynomial ${{x}^{2}}+3x-12$ . Now, we want to divide the polynomial by the linear expression $x-3$ . If we put $x=3$ in the above assumed polynomial, we can see that the polynomial does not become zero. This means that $x=3$ is not a root of the equation ${{x}^{2}}+3x-12=0$ . That means, $x-3$ is not a factor of the polynomial ${{x}^{2}}+3x-12$ . So, ${{x}^{3}}+3x-12$ must give a remainder on division by $x-3$ . If we divide ${{x}^{2}}+3x-12$ by $x-3$ , we get $x+6$ as quotient and $6$ as remainder.
We can write the remainder in the form of a fraction. This is done by at first expression the division as a fraction, that is
$\dfrac{{{x}^{2}}+3x-12}{x-3}$
Since we know that $x+6$ is the quotient and $6$ is the remainder, we can write the polynomial ${{x}^{2}}+3x-12$ in the form $\left( x-3 \right)\left( x+6 \right)+6$ . The fraction thus becomes,
$\begin{align}
  & \Rightarrow \dfrac{\left( x-3 \right)\left( x+6 \right)+6}{x-3} \\
 & =\left( x+6 \right)+\dfrac{6}{x-3} \\
\end{align}$
Therefore, we can say that the remainder upon synthetic division can be shown by a fraction, where the numerator in the remainder and the denominator is the linear expression which is the divisor.

Note:
While expressing the remainder in the fraction form, we should factorise the polynomial correctly, and should remember to add the remainder at the end. The remainder can also be shown in another form. Taking the above example, we can show the synthetic division by $\left( x+6 \right)R6$ where the expression after $R$ is the remainder.
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