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Mastering Polynomial Division: Remainder and Factor Theorems Explained

Learn how to divide polynomials using long division and synthetic division in this comprehensive guide. Understand the Remainder Theorem, which states that when a polynomial ( f(x) ) is divided by ( (x - c) ), the remainder is ( f(c) ). Explore the Factor Theorem that connects factors and zeros of polynomials. Through practical examples, you'll see how to find quotients and remainders, evaluate polynomials, and apply these theorems in solving polynomial equations effectively. Perfect for students and math enthusiasts alike!

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Mastering Polynomial Division: Remainder and Factor Theorems Explained

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  1. Dividing Polynomials;Remainder and Factor Theorems

  2. Objectives • Use long division to divide polynomials. • Use synthetic division to divide polynomials. • Evaluate a polynomials using the Remainder Theorem. • Use the Factor Theorem to solve a polynomial equation.

  3. How do you divide a polynomial by another polynomial? • Perform long division, as you do with numbers! Remember, division is repeated subtraction, so each time you have a new term, you must SUBTRACT it from the previous term. • Work from left to right, starting with the highest degree term. • Just as with numbers, there may be a remainder left. The divisor may not go into the dividend evenly.

  4. Example • Divide using long division. State the quotient, q(x), and the remainder, r(x). (6x³ + 17x² + 27x + 20) (3x + 4)

  5. Example • Divide using long division. State the quotient, q(x), and the remainder, r(x).

  6. Remainders can be useful! • The remainder theorem states: If the polynomial f(x) is divided by (x – c), then the remainder is f(c). • If you can quickly divide, this provides a nice alternative to evaluating f(c).

  7. Quick method of dividing polynomials Used when the divisor is of the form x – c Last column is always the remainder Synthetic Division

  8. Example • Divide using synthetic division.

  9. Example • Divide using synthetic division.

  10. Factor Theorem • f(x) is a polynomial, therefore f(c) = 0 if and only if x – c is a factor of f(x). • Or in other words, • If f(c) = 0, then x – c is a factor of f(x). • If x – c is a factor of f(x), then f(c) = 0. • If we know a factor, we know a zero! • If we know a zero, we know a factor!

  11. Zero of Polynomials If f(x) is a polynomial and if c is a number such that f(c) = 0, then we say that c is a zero of f(x). The following are equivalent ways of saying the same thing. • c is a zero of f(x) • x – c is a factor of f(x)

  12. Example • Use the Remainder Theorem to find the indicated function value.

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