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# Dividing Polynomials; Remainder and Factor Theorems - PowerPoint PPT Presentation

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

• 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.

• 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.

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

(6x³ + 17x² + 27x + 20) (3x + 4)

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

• 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).

Used when the divisor is of the form x – c

Last column is always the remainder

Synthetic Division

• Divide using synthetic division.

• Divide using synthetic division.

• 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!

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)

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