Theory:

In the previous section, we dealt with quadratic equations and their zeroes. However, in this section, we are going to deal with the division algorithm for polynomials.
 
Here, the polynomials have a degree of \(3\) and more.
Division algorithm for polynomials
Let us divide the expression \(x^3 - 3x^2 - 3 + 5x\) by \(x^2 - 2\), try to apply division algorithm on the same, and find the zeroes of the expression step-by-step.
 
Step 1: Arrange the terms in the ascending order of their degrees.
 
The term \(x^3 - 3x^2 - 3 + 5x\) becomes \(x^3 - 3x^2 + 5x - 3\).
 
\(x^2 - 2\) is already in the proper format, and hence it does not require a change.
 
Step 2: Divide the highest degree of the dividend using the highest degree of the divisor.
 
Here, the first term of the dividend is \(x^3\), and the first term of the divisor is \(x^2\).
 
\(\frac{\text{The first term of the dividend}}{\text{The first term of the divisor}} = \frac{x^3}{x^2}\) \(=\) \(x\).
 
Thus, \(x\) becomes the first term of the quotient.
 
1.svg
 
Step 3: Multiply the now-formed quotient and divisor, and subtract from the dividend.
 
\(\text{Quotient} \times \text{Divisor} = x \times (x^2 - 2)\)
 
\(= x^3 - 2x\)
 
Let us now subtract \(x^3 - 2x\) from \(x^3 - 3x^2 + 5x - 3\).
 
2.svg
 
Step 4: Divide the highest degree of the newly-formed dividend using the highest degree of the divisor.
 
Here, the first term of the dividend is \(-3x^2\), and the first term of the divisor is \(x^2\).
 
\(\frac{\text{The first term of the dividend}}{\text{The first term of the divisor}} = \frac{-3x^2}{x^2}\) \(=\) \(-3\).
 
Thus, \(-3\) becomes the second term of the quotient.
 
3.PNG
 
Step 5: Multiply the second term of the quotient with the divisor, and subtract from the dividend.
 
\(\text{Quotient} \times \text{Divisor} = (-3) \times (x^2 - 2)\)
 
\(= -3x^2 + 6\)
 
Let us now subtract \(-3x^2 + 6\) from \(-3x^2 + 7x - 3\).
 
4.PNG
 
Hence, on dividing \(x^3 - 3x^2 - 3 + 5x\) by \(x^2 - 2\), we get \(x - 3\) as the quotient and \(7x - 9\) as the remainder.
 
A dividend is formed by multiplying the quotient with the divisor and then adding the remainder to the result.
 
Irrational pairs of zeroes:
 
In a quadratic equation with rational coefficients has a irrational or surd root \(a + √b\), where \(a\) and \(b\) are rational and \(b\) is not a perfect square, then it has also a conjugate root \(a - √b\).
 
Example:
Consider the quadratic equation \(x^2-2x-1\) with one of its roots \(x = 1+\sqrt2\).
 
Let us find another root by long division method.
 
x(12)x(1+2)x22x1x21+2x()(+)¯12x112x1()(+)¯0
 
Here the other root of the quadratic equation is \(x-(1-\sqrt2) = 0\). That is, \(x = 1-\sqrt2\).
 
Therefore, the other root is \(x = 1-\sqrt2\).
 
It can be noticed that \((1+\sqrt2)\) and \((1-\sqrt2)\) are conjuage pairs.
Important!
Let \(q(x)\), \(d(x)\), and \(r(x)\) be the quotient, divisor and the remainder of the dividend respectively.
 
\(\text{The dividend} = (\text{The quotient} \times \text{The divisor}) + \text{The remainder}\)
 
\(=\) \((q(x)\) \(\times\) \(d(x))\) \(+\) \(r(x)\)