Remainder and Factor Theorem

Unit 11

DefinitionsRoots and Zeros:

The real number, r, is a zero of f(x) iff:1.) r is a solution, or root of f(x)=02.) 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 division2.) Use synthetic division3.) 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 allBook pg 464 #41-45 all