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Section 7.1

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

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  1. 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 = 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

  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. 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. 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 + 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)

  18. 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).

  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 + 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

  21. Section 7.4 Solving Polynomial Equations

  22. 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

  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. • 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

  24. 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

  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. 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.

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