Use long division to divide polynomials We are familiar with the long division algorithm for ordinary arithmetic. How To: Given a polynomial and a binomial, use long division to divide the polynomial by the binomial.
Set up the division problem. Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor.
Multiply the answer by the divisor and write it below the like terms of the dividend. Subtract the bottom binomial from the top binomial. Bring down the next term of the dividend. Repeat steps 2—5 until reaching the last term of the dividend.
If the remainder is non-zero, express as a fraction using the divisor as the denominator. Solution The quotient is [latex]5x - 2. Analysis of the Solution This division problem had a remainder of 0. The remainder is SparkTeach Teacher's Handbook. Summary Long Division of a Polynomial by a Binomial. Long Division of a Polynomial by a Binomial Long division of a polynomial by a binomial is carried out in essentially the same manner as long division of two integers with no variables: Divide the highest degree term of the polynomial by the highest degree term of the binomial.
Previous section Problems Next section Problems. Notice that the leading term is eliminated and that the result has a degree that is one less than the dividend.
The complete process is illustrated below:. Polynomial long division ends when the degree of the remainder is less than the degree of the divisor. Therefore, the binomial divides the polynomial evenly and the answer is the quotient shown above the division line.
To check the answer, multiply the divisor by the quotient to see if you obtain the dividend:. Just as with real numbers, the final answer adds the fraction where the remainder is the numerator and the divisor is the denominator to the quotient. In general, when dividing we have. Since the denominator is a binomial, begin by setting up polynomial long division.
We may write. To check that this result is correct, we multiply as follows:. Occasionally, some of the powers of the variables appear to be missing within a polynomial. This can lead to errors when lining up like terms. Therefore, when first learning how to divide polynomials using long division, fill in the missing terms with zero coefficients, called placeholders.
The division is simplified if we rewrite the expression with placeholders:. Take care to distribute and line up the like terms. Polynomial long division takes time and practice to master. Work lots of problems and remember that you may check your answers by multiplying the quotient by the divisor and adding the remainder if present to obtain the dividend.
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