Synthetic Disvision of Polynomials

Synthetic division is a simplified method for dividing polynomials using only their coefficients. It is used when the divisor is a linear binomial of the form x - c, where c is a real number.

The process begins by writing the dividend's coefficients in descending order of degree, inserting zeros for any missing degrees between the leading term and the constant. The constant c from the divisor is placed on the left side of the division setup.

The first coefficient is written below the line. It is then multiplied by the constant, and the result is added to the next coefficient. This multiply-and-add cycle continues across the row.

The last value in the row represents the remainder, while the preceding values form the coefficients of the quotient—a polynomial with a degree one less than that of the dividend. The final result can be expressed as dividend equals divisor times quotient plus remainder.

In business, polynomials can model profit based on production level. To find the profit at a specific level, the polynomial is divided by a binomial x minus that level.

Using the synthetic division method for this division gives the remainder, which represents the profit at that level.