Divide polynomials using synthetic division

Learning Objectives Identify when synthetic division is an appropriate method for polynomial division. Correctly set up the synthetic division algorithm, including the use of placeholders for missing terms. Execute the 'multiply-and-add' steps of synthetic division accurately. Interpret the final row of numbers to determine the quotient and remainder. Apply the Remainder Theorem to verify the result of a synthetic division problem. Use synthetic division as a tool to test for potential zeros and factors of a polynomial, in accordance with the Factor Theorem. Tired of the long, tedious process of polynomial long division? What if there was a streamlined, elegant shortcut for dividing by linear factors? 🧙‍♂️ This tutorial introduces synthetic division, a highly eff...

TermDefinitionExample DividendThe polynomial that is being divided. It is typically denoted as P(x).In the expression (x³ + 2x² - 5x - 6) ÷ (x - 2), the dividend is x³ + 2x² - 5x - 6. DivisorThe polynomial by which the dividend is divided. For synthetic division, this must be a linear binomial of the form (x - c).In (x³ + 2x² - 5x - 6) ÷ (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.The quotient of (x³ + 2x² - 5x - 6) ÷ (x - 2) is x² + 4x + 3. RemainderThe value left over after the division is complete. If the remainder is 0, the divisor is a factor of the dividend.The remainder of (x³ + 2x² - 5x - 6) ÷ (x - 2) is 0. Zero of a PolynomialA value 'c' such that when substituted into the polyno...

The Division Algorithm for Polynomials P(x) = D(x) \cdot Q(x) + R(x) This rule states that the Dividend (P(x)) is equal to the Divisor (D(x)) multiplied by the Quotient (Q(x)), plus the Remainder (R(x)). This is a fundamental way to check the result of any division problem. The Remainder Theorem If a polynomial P(x) is divided by a linear binomial (x - c), then the remainder is P(c). This theorem provides a powerful shortcut to find the remainder without performing the full division. You can simply evaluate the polynomial at x = c. It's also a great way to check your synthetic division work. The Factor Theorem A polynomial P(x) has a factor (x - c) if and only if P(c) = 0. This is a direct consequence of the Remainder Theorem. If synthetic division by (x - c) yi...