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### Section 7.1

### Section 7.2

### Section 7.3

### Section 7.4

### Section 7.5

An Introduction to Polynomials

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.

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.

Classification of a Polynomial By Degree

Degree Name Example

n = 0 constant 3

n = 1linear 5x + 4

n = 2quadratic-x² + 11x – 5

n = 3cubic 4x³ - x² + 2x – 3

n = 4quartic9x⁴ + 3x³ + 4x² - x + 1

n = 5 quintic-2x⁵ + 3x⁴ - x³ + 3x² - 2x + 6

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

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

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.

Polynomial Functions and Their Graphs

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.

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.

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).

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.

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.

Products and Factors of Polynomials

Multiplying Polynomials

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

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)

Factoring the Sum Difference of Two Cubes

- a³ + b³ = (a + b)(a² - ab + b²)
- a³ - b³ = (a – b)(a² + ab + b²)
- x³ + 27 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)

Factor Theorem and Remainder Theorem

- Factor 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).

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.

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 + 5 2 1 0 1 - 10

x – 2 x³ + 0x²+ x – 10 2 4 10

- (x³ - 2x²) 1 2 5 0

2x² + x

- (2x² - 4x) x² + 2x + 5 is the quotient

5x – 10

- (5x – 10)

0

Solving Polynomial Equations

Use Factoring to Solve

- Solve 3y³ + 9y² - 162y = 0
- 3y³ + 9y² - 162y = 0
- 3y(y² + 3y – 54) = 0
- 3y(y + 9)(y – 6) = 0
- y = 0, - 9, or 6

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.
- 1 1 - 7 15 - 9 (x – 1)(x² - 6x + 9)
- 1 - 6 9 (x – 1)(x – 3)(x – 3)
- 1 - 6 9 0 x = 1 or 3
- The quotient is x² - 6x + 9

Use Variable Substitution

- x⁴ - 4x² + 3 = 0
- (x²)² - 4x² + 3 = 0
- u² - 4u + 3 = 0 (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

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.

Zeros of Polynomial Functions

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

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.

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.

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