Question

If ${x^3} + 1$, $x + 1$, ${x^2} - x + 1$ and $0$ are dividend, divisor, quotient and remainder respectively then verify the division algorithm.

Answer 1

Hint: In this problem, we have to verify the division algorithm. The division algorithm states that for any polynomial $p\left( x \right)$ and $q\left( x \right)$ , there exists unique polynomials $d\left( x \right)$ and $r\left( x \right)$ such that $p\left( x \right) = q\left( x \right) \times d\left( x \right) + r\left( x \right)$ where the value of $r\left( x \right)$ is greater than or equal to $0$ and less than the value of $q\left( x \right)$. We call $p\left( x \right)$ the dividend, $q\left( x \right)$ the divisor, $d\left( x \right)$ the quotient, and $r\left( x \right)$ the remainder. To verify the division algorithm, we will prove that the LHS and RHS of this algorithm are equal.


Complete step-by-step solution: In this problem, it is given that dividend $p\left( x \right) = {x^3} + 1$, divisor $q\left( x \right) = x + 1$, quotient $d\left( x \right) = {x^2} - x + 1$ and remainder $r\left( x \right) = 0$.
The division algorithm states that for any polynomial $p\left( x \right)$ and $q\left( x \right)$ , there exists unique polynomials $d\left( x \right)$ and $r\left( x \right)$ such that $p\left( x \right) = q\left( x \right) \times d\left( x \right) + r\left( x \right)$.
Now first we will write the LHS of the division algorithm. That is, LHS $ = p\left( x \right) = {x^3} + 1$.
Now we are going to find the RHS of the division algorithm. RHS is given by $q\left( x \right) \times d\left( x \right) + r\left( x \right)$.
Therefore, RHS $ = \left( {x + 1} \right) \times \left( {{x^2} - x + 1} \right) + 0$.
Now we are going to simplify the above expression by multiplying two factors. Therefore, we get
RHS $ = x\left( {{x^2} - x + 1} \right) + 1\left( {{x^2} - x + 1} \right)$
$ \Rightarrow $ RHS $ = {x^3} - {x^2} + x + {x^2} - x + 1$
After cancelling the equal terms with opposite signs, we get RHS $ = {x^3} + 1$ which is the same as $p\left( x \right)$ and it is also LHS of the division algorithm in this problem. Therefore, we can say that for given dividend, divisor, quotient and remainder, LHS and RHS of division algorithm are equal.
Therefore, the division algorithm is verified.


Note: The division algorithm states that for any integer $a$ and any positive integer $b$ there exists unique integers $q$ and $r$ such that $a = bq + r$ where $r$ is greater than or equal to $0$ and less than $b$. We call $a$ the dividend, $b$ the divisor, $q$ the quotient and $r$ the remainder. When we are dealing with polynomials, then the operations like addition, subtraction and multiplication are easy to perform, but the division operation is a bit difficult. If we know one factor (divisor) of the polynomial, then we can find other factors by using the division algorithm.