Evaluate polynomials using synthetic division

Learning Objectives State the Remainder Theorem and explain its connection to polynomial evaluation. Correctly set up a synthetic division problem to evaluate a polynomial P(x) at a value x = c. Execute the synthetic division algorithm accurately, including for polynomials with missing terms. Interpret the result of synthetic division to determine the value of P(c). Compare the efficiency of synthetic division versus direct substitution for evaluating higher-degree polynomials. Verify the result of synthetic division by using direct substitution. Ever tried to calculate 3(7)⁵ - 4(7)⁴ + 2(7)³ - 10(7)² + 7 - 20 by hand? 🤯 There's a surprisingly fast and elegant shortcut! This tutorial will teach you how to use synthetic division, a powerful technique for quickly evaluat...

TermDefinitionExample Polynomial in Standard FormA polynomial written with its terms in descending order of exponents. This is crucial for correctly listing coefficients.The polynomial P(x) = 5x - 2x³ + 3 + x⁴ in standard form is P(x) = x⁴ - 2x³ + 0x² + 5x + 3. Direct SubstitutionThe standard method of evaluating a function by replacing the variable with a given number and performing the arithmetic.To evaluate P(x) = x² - 4x + 1 at x = 3, we substitute to get P(3) = (3)² - 4(3) + 1 = 9 - 12 + 1 = -2. Synthetic DivisionA shorthand method for dividing a polynomial by a linear binomial of the form (x - c).Using synthetic division to divide x² - 5x + 6 by (x - 3). The Remainder TheoremA theorem stating that if a polynomial P(x) is divided by (x - c), the remainder of that division is equal to...

The Remainder Theorem If a polynomial P(x) is divided by (x - c), then the remainder R is equal to P(c). This is the foundational theorem that allows us to use a division process (synthetic division) to achieve an evaluation result. The entire technique relies on this principle. The Synthetic Division Algorithm To evaluate P(x) at x = c: \\ 1. Set up: Write 'c' on the left and the coefficients of P(x) on the right. \\ 2. Bring Down: Bring the first coefficient straight down. \\ 3. Multiply & Add: Multiply 'c' by the number you just brought down. Write the result under the next coefficient and add the column. \\ 4. Repeat: Repeat Step 3 until all columns are filled. The last number in the bottom row is the remainder, P(c). This is the step-by-step mech...