Lesson 6-5 and 6-6: Solving Polynomial Equations

Lesson 6-5 and 6-6: Solving Polynomial Equations

Do Now:

1)Find the degree of each polynomial:
a)
b)
c) / 2)Solve the equation:

3)Define each set of numbers:
a)Rational
b)Irrational
c)Imaginary

Consider the equivalent equations and which have -2, 3, and 4 as roots. Notice that all roots are factors of the constant term, 24. In general, if the coefficients (including the constant term) in a polynomial equation are integers, then any root of the equation is a factor of the constant term.

A similar pattern applies to rational roots. Consider the equation, which has roots The numerators 1, 2, and 3 are all factors of the constant term, 6. The denominators 2, 3, and 4 are factors of the leading coefficient, 24.

The Rational Root Theorem
If is in simplest form and is a rational root of the polynomial equation

with integer coefficients, then must be a factor of and must be a factor of .

Steps for finding rational roots of a polynomial equation:

Step 1 – List the possible rational roots.

Step 2 – Substitute the possible root into the equation. If the value is zero, then the number is a root of the polynomial equation.

On the graphing calculator: enter equation in Y1. Then select STAT, EDIT. Enter possible rational roots in L1. Highlight L2, then click on VARS, select Y-Vars, #1, #1. Then in parenthesis, enter L1.

L2=Y1(L1) Enter

Look for 0’s in L2 to determine the actual rational roots.

Example 1 –List the possible rational roots of each polynomial equation. Then find any actual rational roots:

a)
b)

Example 2 – Find all complexroots of each polynomial equation:

a)
b)

Example 3 – Find the irrational roots, if possible:

a)A polynomial equation with rational coefficients has the roots and . Find two additional roots.

b)A polynomial equation with integer coefficients has the roots and . Find two additional roots.

c)A polynomial equation has the root Can you be certain that is a root of the equation? Explain.

Complex Conjugates:______

Example 4 –

a)A polynomial equation with integer coefficients has the roots and Find two additional roots.

b)A student solves a polynomial equation with real coefficients and finds the roots 3, and Explain the student’s mistake.

Example 5 – Write a polynomial equation from the given roots:

a)A third degree polynomial equation with rational coefficients that has roots and / b)A fourth degree polynomial equation with rational coefficients that has roots and .

Example 6 –For the given equation, find the number of complex roots, the possible number of real roots, and the possible number of rational roots.

a) / b)

Example 7 – Find all the zeros of each function:

a) / b)
c) / d)

Example 8 – In each equation, and represent integers. Indicate whether the statement isor true. Explain your answer.

a)A root of the equation is 5.

b)A root of the equation is

c)If is a root of , then is a factor of

d) and are roots of

e) and are roots of

Example 9 –Find a fourth degree polynomial function with real coefficients that has and as zeros and such that

Example 10 – Find a third degree polynomial function with real coefficients that has and as zeros and