dividing polynomials remainder and factor theorems
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Dividing Polynomials; Remainder and Factor Theorems. 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.

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Presentation Transcript
objectives
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.
how do you divide a polynomial by another polynomial
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.
example
Example
  • Divide using long division. State the quotient, q(x), and the remainder, r(x).

(6x³ + 17x² + 27x + 20) (3x + 4)

example1
Example
  • Divide using long division. State the quotient, q(x), and the remainder, r(x).
remainders can be useful
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).
synthetic division
Quick method of dividing polynomials

Used when the divisor is of the form x – c

Last column is always the remainder

Synthetic Division
example2
Example
  • Divide using synthetic division.
example3
Example
  • Divide using synthetic division.
factor theorem
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!
zero of polynomials
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)
example4
Example
  • Use the Remainder Theorem to find the indicated function value.
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