1. An Introduction to Polynomials
Section 7.1
An Introduction to Polynomials

2. Terminology
A monomial is numeral, a variable, or the product of a numeral and one or more values.
Monomials with no variables are called constants.
A coefficient is the numerical factor in a monomial.
The degree of a monomial is the sum of the exponents of its variables.

3. Terminology
A polynomial is a monomial or a sum of terms that are monomials.
Polynomials can be classified by the number of terms they contain.
A polynomial with two terms is binomial. A polynomial with three terms is a trinomial.
The degree of a polynomial is the same as that of its term with the greatest degree.

4. Classification of a Polynomial By Degree
Degree Name Example n = 0 constant 3 n = 1 linear 5x + 4 n = 2 quadratic -x² + 11x – 5 n = 3 cubic 4x³ - x² + 2x – 3 n = 4 quartic 9x⁴ + 3x³ + 4x² - x + 1 n = 5 quintic -2x⁵ + 3x⁴ - x³ + 3x² - 2x + 6

5. Classification of Polynomials
2x³ - 3x + 4x⁵ -2x³ + 3x⁴ + 2x³ + 5
The degree is 5 The degree is 4
Quintic Trinomial Quartic Binomial
x² + 4 – 8x – 2x³ 3x³ + 2 – x³ - 6x⁵
The degree is 3 The degree is 5
Cubic Polynomial Quintic Trinomial

6. Adding and Subtracting Polynomials
The standard form of a polynomial expression is written with the exponents in descending order of degree.
(-2x² - 3x³ + 5x + 4) + (-2x³ + 7x – 6)
- 5x³ - 2x² + 12x – 2
(3x³ - 12x² - 5x + 1) – (-x² + 5x + 8)
(3x³ - 12x² - 5x + 1) + (x² - 5x – 8)
3x³ - 11x² - 10x - 7

7. Graphing Polynomial Functions
A polynomial function is a function that is defined by a polynomial expression.
Graph f(x) = 3x³ - 5x² - 2x +1
Describe its general shape.

8. Polynomial Functions and Their Graphs
Section 7.2
Polynomial Functions and Their Graphs

9. Graphs of Polynomial Functions
When a function rises and then falls over an interval from left to right, the function has a local maximum.
f(a) is a local maximum (plural, local maxima) if there is an interval around a such that f(a) > f(x) for all values of x in the interval, where x ≠ a.
If the function falls and then rises over an interval from left to right, it has a local minimum.
f(a) is a local minimum (plural, local minima) if there is an interval around a such that f(a) < f(x) for all values of x in the interval, where x ≠ a.

10. Graphs of Polynomial Functions
The points on the graph of a polynomial function that correspond to local maxima and local minima are called turning points.
Functions change from increasing to decreasing or from decreasing to increasing at turning points.
A cubic function has at most 2 turning points, and a quartic function has at most 3 turning points. In general, a polynomial function of degree n has at most n – 1 turning points.

11. Increasing and Decreasing Functions
Let x₁ and x₂ be numbers in the domain of a function, f.
The function f is increasing over an open interval if for every x₁ < x₂ in the interval, f(x₁) < f(x₂).
The function f is decreasing over an open interval if for every x₁ < x₂ in the interval, f(x₁) > f(x₂).

12. Continuity of a Polynomial Function
Every polynomial function y = P(x) is continuous for all values of x.
Polynomial functions are one type of continuous functions.
The graph of a continuous function is unbroken.
The graph of a discontinuous function has breaks or holes in it.

13. If a polynomial function is written in standard form
f(x) = a xⁿ + a xⁿ⁻¹ + · · · + a₁x + a₀,
ⁿ ⁿ⁻¹
The leading coefficient is a . ⁿ
The leading coefficient is the coefficient of the term of greatest degree in the polynomial.

14. Products and Factors of Polynomials
Section 7.3
Products and Factors of Polynomials

15. Multiplying Polynomials
x(16 – 2x)(12 – 2x)
x(192 – 32x – 24x + 4x²)
x(192 – 56x + 4x²)
192x – 56x² + 4x³
4x³ - 56x² + 192x

16. Factoring Polynomials
x³ - 5x² - 6x x³ + 4x² + 2x + 8
= x(x² - 5x – 6) = (x³ + 4x²) + (2x + 8)
= x(x – 6)(x + 1) = x²(x + 4) + 2(x + 4)
= (x² + 2)(x + 4)

17. Factoring the Sum Difference of Two Cubes
a³ + b³ = (a + b)(a² - ab + b²)
a³ - b³ = (a – b)(a² + ab + b²)
x³ x³ - 1
= x³ + 3³ = x³ - 1³
= (x + 3)(x² - 3x + 3²) = (x – 1)(x² + 1x + 1²)
= (x + 3)(x² - 3x + 9) = (x – 1)(x² + 1x + 1)

18. Factor Theorem and Remainder Theorem
x – r is a factor of the polynomial expression that defines the function P if and only if r is a solution of P(x) = 0, that is, if and only if P(r) = 0
Remainder Theorem
If the polynomial expression that defines the function of P is divided by x – a, then the remainder is the number P(a).

19. Dividing Polynomials
A polynomial can be divided by a divisor of the form x – r by using long division or a shortened form of long division called synthetic division.
Long division of polynomials is similar to long division of real numbers.

20. Dividing Polynomials
Given that 2 is a zero of P(x) = x³ + x – 10, use division to factor x³ + x – 10.
Use Long Division Use Synthetic Division
x² + 2x
x – 2 x³ + 0x² + x –
- (x³ - 2x²)
2x² + x
- (2x² - 4x) x² + 2x + 5 is the quotient
5x – 10
- (5x – 10)

21. Solving Polynomial Equations
Section 7.4
Solving Polynomial Equations

22. Use Factoring to Solve Solve 3y³ + 9y² - 162y = 0 3y³ + 9y² - 162y = 0
y = 0, - 9, or 6

23. Use a Graph, Synthetic Division, and Factoring to Find All of the Roots of x³ - 7x² + 15x – 9 = 0
x³ - 7x² + 15x – 9 = 0 Use a graph of the related function to approximate the roots. Then use synthetic divisions to test your choices.
(x – 1)(x² - 6x + 9)
(x – 1)(x – 3)(x – 3)
x = 1 or 3
The quotient is x² - 6x + 9

24. Use Variable Substitution
x⁴ - 4x² + 3 = 0
(x²)² - 4x² + 3 = 0
u² - 4u + 3 = (Substitute u in for x²)
(u – 1)(u – 3) = 0
x² = 1 or x² = 3 (Substitute x² in for u)
x = ± √1 or x = ±√3
x = 1, - 1, √3, or - √3

25. Location Principle
If P is a polynomial function and P(x₁) and P(x₂) have opposite signs, then there is a real number r between x₁ and x₂ that is a zero of P, that is, P(r) = 0.

26. Zeros of Polynomial Functions
Section 7.5
Zeros of Polynomial Functions

27. Rational Root Theorem
Let P be a polynomial function with integer coefficients in standard form. If p/q (in lowest terms) is a root of P(x) = 0, then
p is a factor of the constant term of P
q is a factor of the leading coefficient of P

28. Complex Conjugate Root Theorem
If P is a polynomial function with real-number coefficients and a + bi (where b ≠ 0) is a root of P(x) = 0, then a – bi is also a root of P(x) = 0.

29. Fundamental Theorem of Algebra
Every polynomial function of degree n ≥ 1 has at least one complex zero.
Corollary: Every polynomial function of degree n ≥ 1 has exactly n complex zeros, counting multiplicities.