Evaluate polynomials using synthetic division

Learning Objectives Define the Remainder Theorem and explain its connection to polynomial evaluation. Correctly set up a synthetic division problem, including inserting zero coefficients as placeholders for missing terms. Accurately perform the synthetic division algorithm (bring down, multiply, add). Identify the remainder from the result of a synthetic division operation. Interpret the remainder as the value of the polynomial P(x) at x = c. Compare the efficiency of synthetic division versus direct substitution for evaluating polynomials. Apply synthetic division to evaluate polynomials for real, fractional, and complex values of c. Tired of plugging large or complex numbers into long polynomials? 🤔 What if there was a super-fast shortcut that uses only simple addition...

TermDefinitionExample Polynomial in Standard FormA polynomial where the terms are arranged in descending order of their exponents.P(x) = 4x^5 - 7x^3 + 2x - 10. Notice the exponents go 5, 3, 1, 0. CoefficientsThe numerical factors of the terms in a polynomial. When using synthetic division, you must account for all coefficients from the highest degree down to the constant term.For P(x) = 2x^4 - 8x^2 + 5x, the coefficients in order are 2, 0, -8, 5, 0. A zero is used for the missing x^3 term and the constant term. Synthetic DivisionA shorthand method for dividing a polynomial by a linear binomial of the form (x - c).Dividing x^2 + 5x + 6 by x + 2 (where c = -2) can be done quickly using this method. DivisorThe binomial, in the form (x - c), that you are dividing the polynomial by. The value...

The Remainder Theorem If a polynomial P(x) is divided by a linear binomial (x - c), then the remainder R is equal to the value of the polynomial at c, or R = P(c). This is the foundational theorem that allows us to use synthetic division for evaluation. The entire process is designed to find the remainder, which we know is the answer we're looking for. Synthetic Division Setup To evaluate P(x) at x = c: \n1. Write 'c' in a box to the left. \n2. To the right, list the coefficients of P(x) in standard form. Use 0 for any missing terms. \n3. Draw a line below the coefficients. A correct setup is critical for a correct answer. Always double-check that you have the correct sign for 'c' and have included all necessary zero placeholders. The Synthetic D...