Division of Polynomials

Division Terminology

In regular division, \(1430 \div 3 = 276R2\) where 1430 is the dividend, 3 is the divisor, 476 is the quotient, and 2 is the remainder. The division statement can also be written as \(1430 = (3)(276) + 2\) or \(\frac{1430}{3} = 476 + \frac{2}{3}\). In polynomial division, if we take \(f(x)\) as the dividend, \(d(x)\) as the divisor, \(q(x)\) as the quotient and \(r(x)\) as the remainder, we can write the division statement as \(f(x) = d(x)q(x) + r(x)\) and \(\frac{f(x)}{d(x)} = q(x) + \frac{r(x)}{d(x)}\)

Division Methods

Long Division

Treat each term as its own thing, order the terms in the dividend from largest degree to lowest degree and fill in any missing terms with a \(0x^n\). Do the same for the divisor and perform long division.

Synthetic Division

Regular synthetic division can only be used when the divisor is a linear. Modified methods of synthetic division can be used when the divisor is a polynomial. This is too complicated and long division works fine so synthetic division is only recommended for linear divisors.

Key Points

Key things to note for synthetic division are: to divide by the root, not the factor; add, not subtract; when the divisor is in the format of \((ax - b)\), dividing by the root will give the quotient of \((x - \frac{b}{a})\) meaning the quotient still needs to be divided by \(a\) to make the divisor back into \((ax - b)\)

Remainder Theorem

The remainder theorem states that when \(f(x)\) is divided by \((x - a)\) the remainder will be equal to \(f(a)\).