Polynomial Long Division & Synthetic Division

Long Division Algorithm

1. Divide the leading term of the dividend by the leading term of the divisor.
2. Multiply the entire divisor by the result. Write below the dividend, aligned by degree.
3. Subtract to get a new (smaller) polynomial.
4. Bring down the next term. Repeat until the degree of the remainder is less than the degree of the divisor.

Synthetic Division

A shortcut for dividing by a linear factor \((x - c)\). Write the coefficients of the dividend in a row. Bring down the first coefficient, multiply by \(c\), add to the next column, and repeat. The last value is the remainder.

Remainder Theorem

If a polynomial \(P(x)\) is divided by \((x - c)\), the remainder is \(P(c)\).

Factor Theorem

\((x - c)\) is a factor of \(P(x)\) if and only if \(P(c) = 0\).

Rational Root Theorem

If \(P(x) = a_n x^n + \dots + a_0\) has a rational root \(\frac{p}{q}\) in lowest terms, then \(p\) divides \(a_0\) and \(q\) divides \(a_n\).