Chapter 4 Polynomial and Rational Functions
4.1 Polynomial Functions
Which of the following are polynomials? (This includes monomials and binomials)
1
 7𝑥 2 + 5𝑥 − 2



−15𝑥 − √3
4
5𝑥 −1 + 7𝑥 −6 − 𝜋



6𝑥 3 − 10𝑥 4 + 57
3𝑥 2 − 10𝑥10 + 3𝑥 4 − √10𝑦 2 + 5𝑦 4
5
+3
𝑥2

2
1
2𝑥 3 + 5√𝑥 − 10
Ex 1: Consider the following polynomial function 𝑓(𝑥) = 𝑥 3 − 6𝑥 2 + 10𝑥 − 8
a. State the degree and leading coefficient of the polynomial.
b. Determine whether 4 is a zero of 𝑓(𝑥)
How does degree affect the graph of a polynomial??
Fundamental Theorem of Algebra
Ex 2: State the number of real and imaginary solutions
a.
Imaginary/Complex Numbers
Ex 3: Solve the following
a. 7𝑥 2 + 49 = 0
b. 10𝑥 4 + 7𝑥 2 − 12 = 0
c. 𝑥 4 − 8𝑥 2 + 12 = 0
b.
Ex 4: Write an equation whose roots are 3, 4𝑖, −4𝑖
4.2 Quadratic Equations
Complete the Square
Ex 1: Solve 𝑥 2 + 4𝑥 − 17 = 0
Ex 2: Solve 5𝑥 2 − 15𝑥 − 29 = 10
Derive the Quadratic Formula
Ex 3: Find the discriminant of the following and describe the nature of the roots of the equations. Then solve
the equation by using the Quadratic Formula.
a. 𝑥 2 − 4𝑥 + 15 = 0
b. 𝑥 2 + 2𝑥 − 2 = 0
4.3 The Remainder and Factor Theorems
Synthetic Division
Ex 1: Divide 𝑥 3 − 𝑥 2 + 2 by 𝑥 + 1 using synthetic division
Remainder/Factor Theorems
Ex 2: Use the Remainder Theorem to find the remainder for (𝑥 4 + 𝑥 2 + 2) ÷ (𝑥 − 3). State whether the
binomial is a factor of the polynomial.
Ex 3: Find the value of 𝑡 so that (𝑥 3 − 𝑡𝑥 2 + 2𝑥 − 4) ÷ (𝑥 − 2) has the remainder of 0.
4.4 The Rational Root Theorem
Identifying Zeros/Solutions/Roots
Descarte’s Rule of Signs
 Number of sign changes in ________, minus an even number = ________________________________
o Ex:

Number of sign changes in ________, minus an even number = ________________________________
o Ex:
Ex 1: State the number of possible positive and negative real zeros of 𝑓(𝑥) = 5𝑥 4 − 3𝑥 3 + 2𝑥 2 − 7𝑥 + 23
Rational Root Theorem
A way in which we find the POSSIBLE rational real zeros
Ex 2: List the possible rational roots of each equation. Then determine the rational roots.
a. 𝑓(𝑥) = 𝑥 3 + 2𝑥 2 − 5𝑥 − 6
b. 𝑓(𝑥) = 2𝑥 3 + 3𝑥 2 − 𝑥 + 24
4.5 Locating Zeros of a Polynomial Function
Upper Bound/Lower Bound Theorem
Identifies the interval in which ____________________________ are located
Ex 1: Find the integral upper and lower bounds of 𝑓(𝑥) = 𝑥 4 − 3𝑥 3 − 2𝑥 2 + 3𝑥 − 5
Using a Graphing Calculator to Approximate Zeros
Ex 2: Approximate the real zeros of 𝑓(𝑥) = 𝑥 4 − 3𝑥 3 − 2𝑥 2 + 6𝑥 − 13 to the nearest tenth
Ex 3: Use 𝑓(𝑥) = 𝑥 4 − 6𝑥 3 + 2𝑥 2 + 6𝑥 − 13 to
a. Find the integral upper and lower bounds
b. Find the number of positive and negative real zeros
c. Approximate all real zeros to the nearest hundredth
4.7 Radical Equations and Inequalities
Solving Radical Equations
3
Ex 1: Solve √1 − 4𝑡 = 2
Ex 2: Solve √𝑎 + 4 + √𝑎 − 3 = 7
Ex 3: Solve √3𝑥 + 10 = √𝑥 + 11 − 1
Solving Radical Inequalities
Ex 4: Solve 4 ≤ √10𝑥 + 3
Ex 5: Solve 8 > √5𝑥 − 7
Ex 6: Solve √2𝑥 − 9 < −6
4.6 Rational Equations and Partial Fractions
Solving Rational Equations
𝑎2 −5
Ex 1: Solve 𝑎 + 𝑎2 −1 =
𝑎2 +𝑎+2
𝑎+1
Solving Rational Inequalities
1. Ignore_________________. Replace with _________. Solve the ________________________.
2. Find all excluded values (use the original inequality)
3. Put values from #1 and #2 on a __________________________________.
4. Write out the solution based on #3.
5
7
Ex 2: Solve 1 + 𝑥−1 ≤ 6
𝑥 2 −16
Ex 3: Solve 𝑥 2 −4𝑥−5 > 0
Partial Fraction Decomposition
Taking a big fraction and breaking down into smaller fractions
−3𝑥−29
Ex 4: Decompose 𝑥 2 −4𝑥−21
What if A and B are fractions?