8. Polynomial Functions

f(r)

If a polynomial function f(x) is divided by the binomial (x - r), where r is any constant, then the remainder we obtain as a result of this division is the value of f(r)

remainder theorem

If a polynomial function f(x) is divided by the polynomial (x - r), where r is any constant, then the remainder equals f(r).

Synthetic division

If the polynomial we are dividing by is of the form (x - r), the process of dividing one polynomial by another can be simplified by a method called synthetic division. (Using this method, we omit writing the variables and write only the coefficients of the terms, using zero as the coefficient of any missing terms. Terms are written in descending power.)

Degree 3 or more

If you are dealing with polynomials of degree 3 or greater, the theorems in this lesson, used in conjunction with synthetic division, will be very useful in finding the roots of the polynomial.

factor theorem

If, when f(x) = {(a_0 x^n) + (a_1 x^n-1) + (a_2 x^n-2) + . . . + (a_n-1 x) + a_n} is divided by (x - r), we get a remainder of zero; then by the remainder theorem, f(r) = 0. This remainder of zero implies that "r" is a zero of f(x) and that (x - r) is a factor of f(x). Thus, f(x) = (x - r)*q(x) + 0. that r is a zero of f(x) and that (x - r) is a factor of f(x). Thus, f(x) = (x - r) * q(x) + 0.

Converse

The converse of this theorem states that if (x - r) is a factor of f(x), then f(r) = 0.

F(r)=0?

This remainder of zero implies that r is a zero of f(x) and that (x - r) is a factor of f(x). Thus, f(x) = (x - r) * q(x) + 0.

Example 2: Is (x - 3) a factor of f(x) = x^3 - 6x^2 + 11x - 6? Let us find f(3).

f(3) = (3)^3 - 6(3)^2 + 11(3) - 6 = 27 - 54 + 33 - 6 = 60 - 60 = 0 Since f(3) = 0, then (x - 3) is a factor of x^3 - 6x^2 + 11x - 6.

Example 1: f(x) = 5x^3 - 14x + 3 and r = 2

f(r) = 5(2)^3 - 14(2) + 3 = 40 - 28 + 3 remainder = 15 replace x with r=? <----

q(x)

f(x)=q(x)(n-1), degree of q(x) is always 1 less than f(x)

0 as remainder

if you obtain a remainder of 0, then (x - r) must be a factor of the polynomial.

If coefficient is missing

replace with 0

Example

was 1-3, 3 turns positive 1 moves to the first coefficient. Add 3 1-6+11-6 multiply 3 -9 +6 ----------- coefficients > (1-3+2) 0 <-- remainder of 0 (value of f(3))

Rules!!!

1. When f(r) = 0, we know (x - r) is a factor of f(x) 2. If f(x) = polynomial of positive degree f(x) = 2x^3 - 7x^2 + 5x -1 and r = 3 then the remainder is 5 3.Then {f(x) = (2x^3 - 7x^2 + 5x -1) ÷ (x - 3)} = {(2x^2 - x + 2) (x - 3) + 5 } Dividend = Quotient times the Divisor + Remainder

polynomial functions

A group of functions in the form y = a_0 x^n + a_1 x^n-1 + a_2 x^n-2 + ... + a_n-1 x + a_n where a_0, a_1, a_2, . . . are numerical coefficients and n, a positive integer, is the exponent of the variable x. The number n then becomes the degree of the polynomial.