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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Dividing polynomials remainder and factor theorems

Dividing Polynomials;Remainder and Factor Theorems


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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