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Long Division of Polynomials

When attempting to find the roots of a polynomial, it will be useful to be able to divide that polynomial by other polynomials. Here we'll learn how.

Long division of polynomials is a lot like long division of real numbers. If the polynomials involved were written in fraction form, the numerator would be the dividend, and the denominator would be the divisor. To divide polynomials using long division, first divide the first term of the dividend by the first term of the divisor. This is the first term of the quotient. Multiply the new term by the divisor, and subtract this product from the dividend. This difference is the new dividend. Repeat these steps, using the difference as the new dividend until the first term of the divisor is of a greater degree than the new dividend. The last "new dividend" whose degree is less than that of the divisor is the remainder. If the remainder is zero, the divisor divided evenly into the dividend. In the example below, f (x) = x 4 +4x 3 + x - 10 is divided by g(x) = x 2 + 3x - 5 .

Figure %: Long division of polynomials.

Two important theorems pertain to long division of polynomials.

The Remainder Theorem states the following: if a polynomial f (x) is divided by the polynomial g(x) = x - c , then the remainder is the value of f at c , f (c) .

The Factor Theorem states the following: Let f (x) be a polynomial; (x - c) a factor of f if and only if f (c) = 0 . This means that if a given value c is a root of a polynomial, then (x - c) is a factor of that polynomial.

Synthetic division is an easy way to divide polynomials by a polynomial of the form (x - c) . It is both a way to calculate the value of a function at c (Remainder Theorem) as well as to check whether or not c is a root of the polynomial (Factor Theorem). Synthetic division is a shortcut to long division. It requires only three lines -- the top line for the dividend and divisor, the second line for the intermediate values, and the third line for the quotient and remainder. It is done this way. Let the dividend have degree n . 1) In line one write the coefficients of the polynomial as the dividend, and let c be the divisor. 2) In line three rewrite the leading coefficient of the dividend directly below its position in the dividend. 2) Multiply it by the divisor, and write the product in line two directly below the coefficient of x n - 1 . 3) Add this product to the number directly above it in the dividend (this number is the coefficient of x n - 1 ) to get a new number. Repeat steps two and three until the entire polynomial has been divided. The quotient will be one degree less than the dividend. The coefficients of the quotient are the first n - 1 numbers in line three. The remainder is the last number in line three. Below a polynomial of the form (x - c) is divided using long division, and then using synthetic division. Study it carefully.

Figure %: Long division of a polynomial of the form (x - c) .
Figure %: Synthetic division.

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