21 = P(x) = Q(x) . (x - r) + P(r) 4.3 The Remainder Theorem

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Presentation transcript:

21 = 4 . 5 + 1 P(x) = Q(x) . (x - r) + P(r) 4.3 The Remainder Theorem dividend=quotient . divisor + remainder If a polynomial P(x) is divided by x - r, the remainder is the constant P(r), and P(x) = Q(x) . (x - r) + P(r) where Q(x) is a polynomial with degree one less than the degree of P(x).

Example Problem 1 Find the remainder of divided by x + 1 Use the Remainder Theorem!!!

Synthetic Division

The Factor Theorem The binomial x - r is a factor of the polynomial P(x) if and only if P(r) = 0. Example Problem 2 Use the factor theorem to determine whether the function has the factor x – 1. See if f(1) = 0, cause if it does then x – 1 is a factor

Depressed Polynomial When a polynomial is divided by one of its binomial factors x – r, the quotient is called a depressed polynomial. x – 1 is a factor of 2x3 – 3x2 + x. Using synthetic division you can find the depressed polynomial to be 2x2 – x.

Review Example 4 & 5 on pg. 225 Take note of the graphing calculator tip The remainder theorem can be used to determine missing coefficients