Divide polynomials using synthetic division

Learning Objectives Identify when synthetic division is an appropriate method for dividing polynomials. Correctly set up a synthetic division problem, including the use of placeholders for missing terms. Execute the synthetic division algorithm accurately to find the quotient and remainder. Interpret the results of synthetic division to write the final answer in the form Q(x) + R/(x-c). Apply the Remainder Theorem to evaluate a polynomial for a given value. Use the Factor Theorem to determine if a linear binomial is a factor of a polynomial. Tired of the long, tedious process of polynomial long division? 🤔 What if there was a shortcut that was faster and less prone to error? This tutorial introduces synthetic division, a streamlined method for dividing a polynomial by a li...

TermDefinitionExample DividendThe polynomial that is being divided. It is typically of a higher degree than the divisor.In the problem (3x³ - 2x² + 4x - 9) ÷ (x - 2), the dividend is 3x³ - 2x² + 4x - 9. DivisorThe polynomial by which the dividend is divided. For synthetic division, this must be a linear binomial of the form (x - c).In the problem (3x³ - 2x² + 4x - 9) ÷ (x - 2), the divisor is (x - 2). QuotientThe main result of the division. Its degree is always one less than the degree of the dividend when using synthetic division.When (x² - 5x + 6) is divided by (x - 2), the quotient is (x - 3). RemainderThe value left over after the division is complete. If the remainder is zero, the divisor is a factor of the dividend.When (x² - 5x + 7) is divided by (x - 2), the remainder is 1. Root...

Synthetic Division Setup To divide P(x) by (x - c), set up a bracket with 'c' on the outside and the coefficients of P(x) on the inside. Remember to use a zero placeholder for any missing terms. This is the fundamental structure for every synthetic division problem. The value 'c' is the root of the divisor. For a divisor like (x + 5), 'c' would be -5. The Remainder Theorem If a polynomial P(x) is divided by (x - c), then the remainder is equal to P(c). This theorem provides a powerful shortcut. Instead of substituting 'c' into a complex polynomial, you can use synthetic division. The last number in your result is the answer. The Factor Theorem (x - c) is a factor of a polynomial P(x) if and only if P(c) = 0. This is a direct co...