Presentation on theme: "Remainder and Factor Theorem Unit 11. Definitions Roots and Zeros: The real number, r, is a zero of f(x) iff: 1.) r is a solution, or root of f(x)=0 2.)"— Presentation transcript:
Definitions Roots and Zeros: The real number, r, is a zero of f(x) iff: 1.) r is a solution, or root of f(x)=0 2.) x – r is a factor of the expression that defines f (that is, f(r) =0 ) 3.) When the expression is divided by x – r, the remainder is 0. 4.) r is an x-intercept of the graph of f.
Factor Theorem O x – r is a factor of the polynomial expression that defines the function P iff r is a solution of P(x)=0. That is if P(r)=0. O Using the Factor Theorem, you can test for linear factors involving integers by using substitution.
Examples: Use substitution to determine whether x+3 is a factor of Use substitution to determine if x - 1 is a factor of
Basically: To check if P(r)=0, you can: 1.) Use long division 2.) Use synthetic division 3.) Substitute r into the expression to see if P(r)=0.
Remainder Theorem O If the polynomial expression that defines the function of P is divided by x – r, then the remainder is the number P(r).
Examples O Given find P(5) *If the result is zero, then x – 5 is a factor. *If the result is any other value, that value is the remainder.
Examples O Given find P(3) *You can check your work by using synthetic division.
Application: In order to pull it all together, we can work backwards to form the function, P given the degree of P and the zeros. Start with the factored form and P(0) to find the lead coefficient, a.
Example Write a polynomial function, P, in factored and standard form using the given information: 1.) P is degree 2, P(0) =12; zeros: 2, 3 2.) P is degree 4, P(0)=1, zeros; 1 (multiplicity 2) and 2 (multiplicity 2)
Assignment: Book pg 446 #61-71 all and 91-98 all Book pg 464 #41-45 all