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Section 7.1. 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.

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Section 7 1

Section 7.1

An Introduction to Polynomials


Terminology
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.


Terminology1
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
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
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
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
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.


Section 7 2

Section 7.2

Polynomial Functions and Their Graphs


Graphs of polynomial functions
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 functions1
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
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
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
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 .

    The leading coefficient is the coefficient of the term of greatest degree in the polynomial.


Section 7 3

Section 7.3

Products and Factors of Polynomials


Multiplying 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
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
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 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
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 polynomials1
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


Section 7 4

Section 7.4

Solving Polynomial Equations


Use factoring to solve
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 d ivision and factoring to find all of the roots of x 7x 15x 9 0
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
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
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.


Section 7 5

Section 7.5

Zeros of Polynomial Functions


Rational root theorem
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
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
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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