Lesson 7: 4-9 Roots and Zeros

1. Suppose that you have a list of all the zeros {x1, x2, x3}, of a polynomial function, p(x). What is the degree of p(x)?
2. Suppose that you have a list of all the zeros, {x1, x2, x3, x4, x5}, of a polynomial function, p(x). What are the factors of p(x)?

Roots
Zeros
Intercepts
Factors
Fundamental Theorem of Algebra
Descartes' Rule of Signs
Complex Conjugate

N.CN.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

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 real zeros of a polynomial function f(x) are the x-intercepts of the graph of f(x). They are also the real solutions of the polynomial equation f(x)=0. A polynomial function cannot have more zeros than its degree.

The function g(x)=0.002x^4-0.049x^3+0.404x^2-1.00x+1.979 can be used to model the average price of a gallon of gasoline in a given year if x is the number of years since 2000. To find the average price of gasoline in a specific year, you can use the roots of the related polynomial equation.

Point out to students the method for determining the number of imaginary zeros for a polynomial function. For example, a polynomial that has a degree of 6 has a maximum of 6 real zeros. You find the number of positive and negative real zeros, and subtract the sum of these two numbers from 6 to find the number of imaginary zeros. Remind students that imaginary zeros come in conjugate pairs, so the number of imaginary zeros must be an even number.

Stress that throughout this course, students must work using neat and careful handwriting. It is extremely easy to misread coefficients and exponents, or misread i as the number 1.

McGraw Hill resources

McGraw Hill resources