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.

• 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 3n = 1 linear 5x + 4n = 2 quadratic -x² + 11x – 5n = 3 cubic 4x³ - x² + 2x – 3 n = 4 quartic 9x⁴ + 3x³ + 4x² - x + 1n = 5 quintic -2x⁵ + 3x⁴ - x³ + 3x² - 2x + 6

Classification of Polynomials

• 2x - 3x + 4x³ ⁵ -2x + 3x + 2x + ³ ⁴ ³5• The degree is 5 The degree is 4• Quintic TrinomialQuartic Binomial• x + 4 – 8x – 2x² ³ 3x + 2 – x - 6x³ ³ ⁵• 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.

Section 7.2

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 + · · · + a x + a ,⁻ ₁ ₀ ⁿ ⁿ ¹⁻The leading coefficient is a . ⁿ

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

Section 7.3

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

Section 7.4

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.

Section 7.5

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.