Lesson 6-5 and 6-6: Solving Polynomial Equations
Do Now:
1) Find the degree of each polynomial:
a) 3𝑥 2 − 𝑥 + 5
2) Solve the equation:
2𝑥 2 + 5𝑥 = −4
b) – 𝑥 + 3 − 𝑥
3
c) −4𝑥 5 + 1
3) Define each set of numbers:
a) Rational
b) Irrational
c) Imaginary
Consider the equivalent equations 𝑥 3 − 5𝑥 2 − 2𝑥 + 24 = 0 and (𝑥 + 2)(𝑥 − 3)(𝑥 − 4) = 0, 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 24𝑥 3 − 22𝑥 2 − 5𝑥 + 6 = 0, which has roots
1 2
3
− 2 , 3 , 𝑎𝑛𝑑 4. 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
𝑝
If 𝑞 is in simplest form and is a rational root of the polynomial equation
𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 + 𝑎0 = 0
with integer coefficients, then 𝑝 must be a factor of 𝑎0 and 𝑞 must be a factor of 𝑎𝑛 .
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) 𝑥 3 + 𝑥 2 − 3𝑥 − 3 = 0
b) 12𝑥 3 − 32𝑥 2 + 25𝑥 − 6 = 0
Example 2 – Find all complex roots of each polynomial equation:
a) 2𝑥 3 − 𝑥 2 + 2𝑥 − 1 = 0
b) 5𝑥 3 − 24𝑥 2 + 41𝑥 − 20 = 0
The Irrational Root Theorem
Let 𝑎 and 𝑏 be rational numbers and let √𝑏 be an irrational number. If 𝑎 + √𝑏 is a root of a polynomial
equation with rational coefficients, then the conjugate 𝑎 − √𝑏 is also a root. (Note: this theorem applies to
even roots. Odd roots do not have a conjugate. Therefore, it is possible for a problem to have an odd number
of irrational solutions.)
Example 3 – Find the irrational roots, if possible:
a) A polynomial equation with rational coefficients has the roots 2 − √7 and √5. Find two additional roots.
b) A polynomial equation with integer coefficients has the roots 1 + √3 and −√11. Find two additional
roots.
c) A polynomial equation has the root 4 − √2. Can you be certain that 4 + √2 is a root of the equation?
Explain.
Complex Conjugates:_________________________________________
Imaginary Root Theorem
If the imaginary number 𝑎 + 𝑏𝑖 is a root of a polynomial equation with real coefficients, then the conjugate
𝑎 − 𝑏𝑖 is also a root.
Example 4 –
a) A polynomial equation with integer coefficients has the roots 3 − 𝑖 and 2𝑖. Find two additional roots.
b) A student solves a polynomial equation with real coefficients and finds the roots 3, 1 − √2 and 4𝑖.
Explain the student’s mistake.
Fundamental Theorem of Algebra
If 𝑃(𝑥) is a polynomial of degree 𝑛 ≥ 1 with complex coefficients, then 𝑃(𝑥) = 0 has at least one complex
root.
Corollary
Including imaginary roots and multiple roots, an 𝑛th degree polynomial equation has exactly 𝑛 roots; the
related polynomial function has 𝑛 zeros.
Example 5 – Write a polynomial equation from the given roots:
a) A third degree polynomial equation with rational
coefficients that has roots 3 and 1 + 𝑖.
b) A fourth degree polynomial equation with rational
coefficients that has roots 3 + √2 and √5.
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) 𝑥 3 + 2𝑥 2 − 4𝑥 − 6 = 0
b) 𝑥 4 − 3𝑥 3 + 𝑥 2 − 𝑥 + 3 = 0
Example 7 – Find all the zeros of each function:
a) 𝑓(𝑥) = 𝑥 3 + 𝑥 2 − 𝑥 + 2
b) 𝑦 = 𝑥 5 + 3𝑥 4 − 𝑥 − 3
c) 𝑓(𝑥) = 2𝑥 4 + 12𝑥 3 − 𝑥 2 − 7𝑥 − 6
d) 𝑦 = 4𝑥 4 − 𝑥 2 + 2𝑥 − 1
Example 8 – In each equation, 𝑟, 𝑠, and 𝑡 represent integers. Indicate whether the statement is
𝑠𝑜𝑚𝑒𝑡𝑖𝑚𝑒𝑠, 𝑎𝑙𝑤𝑎𝑦𝑠, or 𝑛𝑒𝑣𝑒𝑟 true. Explain your answer.
a) A root of the equation 3𝑥 3 + 𝑟𝑥 2 + 𝑠𝑥 + 8 = 0 is 5.
b) A root of the equation 3𝑥 3 + 𝑟𝑥 2 + 𝑠𝑥 + 8 = 0 is −2.
c) If 𝑎 is a root of 𝑥 3 + 𝑟𝑥 2 + 𝑠𝑥 + 𝑡 = 0, then 𝑎 is a factor of 𝑡.
d) √5 and −√5 are roots of 𝑥 3 + 𝑟𝑥 2 + 𝑠𝑥 + 𝑡 = 0.
e) 2 + 𝑖 and −2 − 𝑖 are roots of 𝑥 3 + 𝑟𝑥 2 + 𝑠𝑥 + 𝑡 = 0.
Example 9 – Find a fourth degree polynomial function 𝑓(𝑥) with real coefficients that has −2, 2, and 𝑖 as zeros
and such that 𝑓(3) = −150.
Example 10 – Find a third degree polynomial function 𝑓(𝑥) with real coefficients that has −5 and 4 + 3𝑖 as
zeros and 𝑓(2) = 91.
Homework:
Page 333 #’s 1-6, 8, 10, 12, 25, 26, and 28 (examples 1 and 2), #’s 13-18 (examples 3 and 4), 19-24 and 30 (example 5).
Page 337 #’s 2-16 even.