Roots & Zeros of Polynomials I How the roots, solutions, zeros, x-intercepts and factors of a polynomial function are related.

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Presentation transcript:

Roots & Zeros of Polynomials I How the roots, solutions, zeros, x-intercepts and factors of a polynomial function are related.

Polynomials A Polynomial Expression can be a monomial or a sum of monomials. The Polynomial Expressions that we are discussing today are in terms of one variable. In a Polynomial Equation, two polynomials are set equal to each other.

Factoring Polynomials Terms are Factors of a Polynomial if, when they are multiplied, they equal that polynomial: (x - 3) and (x + 5) are Factors of the polynomial

Since Factors are a Product... …and the only way a product can equal zero is if one or more of the factors are zero… …then the only way the polynomial can equal zero is if one or more of the factors are zero.

Solving a Polynomial Equation The only way that x 2 +2x - 15 can = 0 is if x = -5 or x = 3 Rearrange the terms to have zero on one side: Factor: Set each factor equal to zero and solve:

Solutions/Roots a Polynomial Setting the Factors of a Polynomial Expression equal to zero gives the Solutions to the Equation when the polynomial expression equals zero. Another name for the Solutions of a Polynomial is the Roots of a Polynomial !

Zeros of a Polynomial Function A Polynomial Function is usually written in function notation or in terms of x and y. The Zeros of a Polynomial Function are the solutions to the equation you get when you set the polynomial equal to zero.

Zeros of a Polynomial Function The Zeros of a Polynomial Function ARE the Solutions to the Polynomial Equation when the polynomial equals zero.

Graph of a Polynomial Function Here is the graph of our polynomial function: The Zeros of the Polynomial are the values of x when the polynomial equals zero. In other words, the Zeros are the x-values where y equals zero.

x-Intercepts of a Polynomial The points where y = 0 are called the x-intercepts of the graph. The x-intercepts for our graph are the points... and (-5, 0) (3, 0)

x-Intercepts of a Polynomial When the Factors of a Polynomial Expression are set equal to zero, we get the Solutions or Roots of the Polynomial Equation. The Solutions/Roots of the Polynomial Equation are the x-coordinates for the x-Intercepts of the Polynomial Graph!

Factors, Roots, Zeros For our Polynomial Function: The Factors are:(x + 5) & (x - 3) The Roots/Solutions are:x = -5 and 3 The Zeros are at:(-5, 0) and (3, 0)