Lesson 6: 4-8 The Remainder and Factor Theorem
Duration of Days: 2
Lesson Objective
1. Evaluate functions by using synthetic substitution.
2. Determine whether a binomial is a factor of a polynomial by using synthetic substitution.
1. In division, what does a remainder of zero tell you?
2. In synthetic division, how would you write the quotient when the remainder is R?
3. If x-r is a factor of polynomial P(x), how does that help you rewrite P(x) in factored form?
4. If P(x)=0 and x-r is a factor of polynomial P(x), what is P(r)? Support your answer.
Synthetic Substitution
Depressed Polynomial
A.APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x-a is p(a), so p(a)=0 if and only if (x-a) is a factor of p(x).
F.IF.7.c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
SAT questions related to polynomial operations: 7-3-2, 6-4-1, 4-3-5, 3-4-6, 1-3-5, 4-4-2, 3-4-33, 8-3-5; dividing polynomials: 7-3-13, 2-3-15
The Remainder Theorem says that the value of f(a) is the same as the remainder when the polynomial is divided by x-a. The Factor Theorem is a special case of the Remainder Theorem. It says: If f(a) has a value of 0, then x-a is a factor of the polynomial.
The number of college students from the United States who study abroad can be modeled by the function S(x)=0.02x^4-0.52x^3+4.03x^2+0.09x+77.54, where x is the number of years since 1993 and S(x) is the number of students in thousands. You can use this function to estimate the number of U.S. college students studying abroad in 2020 by evaluating the function for x=27. Another method you can use is synthetic substitution.
Point out that the Factor Theorem does not say anything about which numbers to try. Techniques for identifying potential factors will be introduced later in the chapter.
Remind students that not all polynomials can be factored. Emphasize that the factors indicate where the graph of the function crosses the x-axis. If the graph of a polynomial function has no x-intercepts, then the polynomial cannot be factored.
Have students write explanations that will help them review the material later.
McGraw Hill resources
McGraw Hill resources