Question

State division algorithm for polynomials.

Answer 1

Hint: If in a polynomial the variable is \[x\] then we can represent that polynomial as a function of \[x\]. It means the polynomial can be expressed as \[f\left( x \right)\] or \[g\left( x \right)\]. The division algorithm for polynomials is identical as the division rule which we use for the numbers which states that:
Dividend \[ = \] divisor \[ \times \] quotient \[ + \] remainder

Complete step-by-step solution:
Let we have two polynomials \[f\left( x \right)\] and \[g\left( x \right)\], where degree of \[f\left( x \right)\] is greater than that of \[g\left( x \right)\] where \[g\left( x \right) \ne 0\].
By degree of a polynomial, we mean the highest power of the variable in that polynomial. For example, let we have a polynomial in variable \[x\]:
\[{x^3} - 4{x^2} + 6x + 2\]
Here we can see that the highest power of \[x\] is \[3\], so the degree of the polynomial is \[3\].
The division algorithm for polynomial states that:
“If \[f\left( x \right)\] and \[g\left( x \right)\] are two polynomials such that degree of \[f\left( x \right)\] is greater than degree of \[g\left( x \right)\] where \[g\left( x \right) \ne 0\], then there exists unique polynomials \[q\left( x \right)\] and \[r\left( x \right)\] such that :
\[f\left( x \right) = q\left( x \right) \times g\left( x \right) + r\left( x \right)\],
Where, \[r\left( x \right) = 0\] or degree of \[r\left( x \right)\] is less than degree of \[g\left( x \right)\].”
We can see that the division algorithm for polynomials is identical to that of the division rule for numbers. Here,
\[f\left( x \right)\] is dividend
\[g\left( x \right)\] is divisor
\[q\left( x \right)\] is quotient
\[r\left( x \right)\] is the remainder.
For finding \[q\left( x \right)\] and \[r\left( x \right)\] we will divide \[f\left( x \right)\] by \[g\left( x \right)\].
For division we follow the following rules:
First, we will divide the highest degree term of the dividend by the highest degree term of the divisor and obtain the remainder.
Then if the remainder is \[0\] or if the degree of the remainder is less than that of the divisor, then we will stop further division and if the degree of the remainder is equal to or greater than that of the divisor then we will do further division.

Note: One important point to note here is that the divisor that is \[g\left( x \right)\] should not be equal to \[0\] because division by zero is meaningless. Another point to note is that the degree of divisor should be less than that of dividend.