Zeros of Polynomials

Notes:

Terms

Rational Suspects: The factors we’ve found based on the polynomial function.

Real-Number Zero: A number that when plugged into the polynomial function; Returns 0 in the remainder.

Example Problem to reinforce the idea

Suppose that x = (5⁄3) is a zero of a polynomial function that has INTEGER coefficients. What does that mean about a factor of the polynomial?

x = (5/3)

can also be…

5x = 3

Or even…

5x - 3 = 0

Hence… 5x-3 is a factor of the polynomial!

So for… P(x) = (5x-3)Q(x), - Leading term of P(x) is 5x. - The leading coefficient of P(x) is a multiple of 5.

  • Constant term of P(x) is -3.
    • The constant term of P(x) is a multiple of 3.

Theorems

Let P(x) be a polynomial with integer coefficients {

  Where 'a' is a factor of the '**Constant Term**', and 'b' is a factor of the '**Leading Coefficient**' of P(x) {

  Any rational number zeros of P(x) are of the form: `x = (a/b)`;

  }

}