Exponential LAWS Review NO CALCULATORS Name: Operations w/polynomials 4.0 Class: Topic: Operations with Polynomials Date: Main Ideas: Assignment: Given: f(x) = x 2 6x 9 a) Find the y-intercept, the equation of axis of symmetry, and coordinates of the vertex. b) Make a table of values that includes the vertex in the middle and 4 other points include all key points in table and have them symmetric about the vertex. c) Use this information to graph the quadratic. LAW Algebraic Meaning Example Product of Powers x a x b = x a+b x 2 y 4 x 5 y = Quotient of Powers {x 0} x a = xa b xb 4x 7 y 8 2x 5 y = Negative Exponents {x 0} x a = 1 x a and 1 = xa x a b 5 = 1 h 9 = Power of a Power (x a ) b = x ab (v 6 ) 7 = Power of a Product (xy) a = x a y a (x 5 y 7 ) 3 = Power of a Quotient {x 0, y 0} Zero Power {x 0} ( x y ) a = xa y a ( x a y ) = ( y a x ) x 0 = 1 = ya x a ( x5 y 7 3 xy 5 ) = ( (x4 y 5 ) 4 0 (xy 5 ) 3 ) =
Other Examples Degree and Naming Polynomials Simplifying Monomials Simplify using no negative exponents. (8x 3 y 3 )( 2x 5 y 6 ) 3 18c 7 d 5 3c 2 d 7 ( 3a 4 ) 3 ( 2x 3 y 2 ) 5 Definition of a polynomial is a function in the form f(x) = a n x n + a n 1 x n 1 + + a 1 x + a 0 a n 0, All exponents are whole # s and all coefficients are real # s a n is the Leading Coefficient (or LC) n is the Degree of the polynomial a 0 is the constant term f(x) = 1 4 x4 8x 5 f(x) = 2x 3 5x 4 + 2 f(x) = x + x + 4 f(x) = x 3 f(x) = x 3 + 2x 2 + 6 f(x) = x 7 6x 8 ( x 2 3x + 4) (x 2 + 2x + 5) (x + 7)(2x 2 + 3x 5) (x 2 5x + 16)(x 2 3x 5) (x 5)(x 3)(x + 4)
Name: Dividing Polynomials 4.1 Class: Topic: Dividing Polynomials Date: Main Ideas: Assignment: b 2 b 5 b 3 6ac 4 3a 2 c 2 Review NO CALCULATORS (10a 2 6ab + b 2 ) (5a 2 2b 2 ) 3y(2y 2 1)(y + 4) Dividing a Polynomial by a Monomial 5a 2 b 15ab 3 + 10a 3 b 4 5ab 3x 2 y + 6x 5 y 2 9x 7 y 3 3x 2 y 204 17 (x 2 2x 15) (x 5) Long Division
(x 2 + 5x + 6) (x + 3) (x 2 7x + 12) (x 4) Remainders for Understanding (a 2 5a + 3)(2 a) 1 (x 2 x 7)(x 3) 1 (x 4 4x 3 7x 2 + 34x 24) (x 2 5x + 4) Upper Level (x 4 3x 3 3x 2 + 7x + 6) (x 2 + 2x + 1)
Dividing Polynomials 4.1 (x 3 4x 2 + 6x 4) (x 2) Synthetic Division (Steps on Page 314) (g 4 3g 2 18) (g 2) (x 2 + 8x + 7) (x + 1) (4y 3 6y 2 + 4y 1) (2y 1) Divisor with LC other than 1 (8y 3 12y 2 + 4y + 10) (2y + 1)
Name: Polynomial Functions 4.2 Class: Topic: Polynomial Functions Date: Main Ideas: Review Assignment: The volume of a box is given by the expression x 3 + 3x 2 x 3. The height of the box is given by the expression (x 1). Find an expression for the area of the base of the box. Then find the expressions that represent the dimensions of the base. Degree and LC State the degree and leading coefficient of each polynomial. If it is not a polynomial in one variable, explain why. 7z 3 4z 2 + z 6a 3 4a 2 + ab 2 3x 5 + 2x 2 4 8x 6 9y 3 + 4y 6 45 8y 2 5y 7 Function Values Upper Level Find f(3)when f(x) = 2x 2 8x + 7 Find f(x 5)when f(x) = 2x 2 8x + 7 Find b(2x 1) 3b(x) if b(m) = 2m 2 + m 1 Find f(m)when f(x) = 2x 2 8x + 7 Find 5f(x)when f(x) = 2x 2 8x + 7 Find g(2x + 1) 2g(x) if g(b) = b 2 + 3
Key Features of Polynomial Functions x -3-2 -1 0 1 2 3 f(x) f(x) = x 3 + x 2 5x + 1 Vocabulary: Location Principal if the value of f(x) changes signs from one value of x to the next, then there is a zero (x-intercept, root) between the two x-values. Local Extrema: Relative Maximum the tops of the mountains (peaks) are relative maximums because they are the highest points in their little neighborhood Relative Minimum the lowest points (valleys) in the little neighborhood are relative minimums Turning Points Relative MAX and MIN are often referred to as turning points. A polynomial function with a degree n has at most (n-1) turning points Constant Function Degree 0 Linear Function Degree 1 Quadratic Function Degree 2 Graphs of Polynomial Functions Cubic Function Degree 3 Quartic Function Degree 4 Quintic Function Degree 5
Polynomial Functions 4.2 Degree: even LC: positive End Behaviors: as x f(x) + as x + f(x) + Degree: odd LC: positive End Behaviors: as x f(x) as x + f(x) + End Behaviors Domain: all real # s Range: all real numbers minimum Degree: even LC: negative End Behaviors: as x f(x) Domain: all real # s Range: all real numbers Degree: odd LC: negative End Behaviors: as x + f(x) as x + f(x) as x + f(x) Domain: all real # s Range: all real numbers maximum Domain: all real # s Range: all real numbers Odd Degree Functions will always have an odd number of real zeros. Even Degree Functions will always have an even number of real zeros or no real zeros at all. Even and Odd Degree
Examples For each graph: a) Describe the end behaviors, b) Determine whether it represents an odd-degree or an even degree polynomial function, and c) State the number of real zeros.
Name: Remainder and Factor 4.3 Class: Topic: Remainder and Factor Theorems Date: Main Ideas: Assignment: Factor 8c 3 g 3 Factor 12az 6bz 6cz + 10ax 5bx 5cx Solve 16d 4 48d 2 + 32 = 0 (all zeros) Factor 8x 3 m 2 8x 3 n 2 + y 3 m 2 y 3 n 2 Review Solve k 3 + 64 = 0 (all zeros) The width of a box is 3 feet less than the length. The height is 4 feet less than the length. The volume of the box is 36 cubic feet. Find the length of the box. Words: If polynomial P(x) is divided by (x r), the remainder is a constant P(r), and Remainder Theorem Dividend equals qoutient times divisor plus remainder P(x) = Q(x) (x r) + P(r), where Q(x) is a polynomial with degree one less than P(x). Example: x 2 + 6x + 2 = (x 4) (x + 10) + 42 Deeper Look Synthetic Division vs. Synthetic/Direct Substitution Synthetic Division (2x 4 5x 2 + 8x 7) (x 6) Synthetic/Direct Substitution Find f(6)for f(x) = 2x 4 5x 2 + 8x 7
If f(x) = 2x 3 3x 2 + 7, find f(3) Factor Theorem Your Turn Words: The binomial (x r) is a factor of the polynomial P(x) if and only if P(r) = 0. Examples: (x 2 2x 15) (x 5) Determine whether x 3 is a factor of x 3 + 4x 2 15x 18. Then find the remaining factors of the polynomial. Examples (Depressed Polynomial) Determine whether x + 2 is a factor of x 3 + 8x 2 + 17x + 10. If so, find the remaining factors of the polynomial.
Name: Roots and Zeros 4.4/4.5 Class: Topic: Roots and Zeros Date: Main Ideas: Assignment: Use synthetic substitution to find f(2) for f(r) = 3r 4 + 7r 2 12r + 23 Use synthetic substitution to find f(6) for f(c) = 2c 3 + 19c 2 + 2 Review Given a polynomial and one of its factors, find the remaining factors of the polynomial. k 4 + 7k 3 + 9k 2 7k 10; (k + 2) Application Question The function f(x) = x 3 6x 2 x + 30 can be used to describe the relative stability of a small boat carrying x passengers, where f(x) = 0 indicates that the boat is extremely unstable. With three passengers, the boat tends to capsize. What other passenger loads could cause the boat to capsize? Zeros, Roots, Factors, Intercepts, Solutions Words: Let P(x) = a n x n + + a 1 x + a 0 be a polynomial function. Then the following statements are equivalent. c is a zero of P(x). c is a root or solution of P(x) = 0. x c is a factor of a n x n + + a 1 x + a 0. If c is a real number, then (c, 0) is an x-intercept of the graph of P(x). Example: Consider the polynomial function P(x) = x 4 + 2x 3 7x 2 8x + 12. The zeros of P(x) = x 4 + 2x 3 7x 2 8x + 12 are 3, 2, 1, and 2. The roots of x 4 + 2x 3 7x 2 8x + 12 = 0 are 3, 2, 1, and 2. The factors of x 4 + 2x 3 7x 2 8x + 12 are (x + 3), (x + 2), (x 1), and (x 2). The x-intercepts of the graph of P(x) = x 4 + 2x 3 7x 2 8x + 12 are ( 3, 0), ( 2, 0), (1, 0), and (2, 0).
Fundamental Theorem of Algebra Every polynomial equation with a degree greater than zero has at least one root in the set of complex numbers. Example: Solve x 2 + 2x 48 = 0. State the number and type of roots. Solve y 4 256 = 0. State the number and types of roots. Your Turn Solve x 2 x 12 = 0. State the number and type of roots. Solve a 4 81 = 0. State the number and type of roots. Corollary to the Fundamental Theorem A polynomial equation with degree n has exactly n roots in the set of complex numbers, including repeated roots. Examples: x 3 + 2x 2 + 6 4x 4 3x 3 + 5x 6 2x 5 3x 2 + 8 Repeated Roots State the number and type of roots. f(x) = x 2 + 6x + 9
Descartes Rule of Signs Roots and Zeros 4.4/4.5 Let P(x) = a n x n + a 1 x + a 0 be a polynomial function with real coefficients. Then: the number of positive zeros of P(x) is the same as the number of changes in signs of the coefficients of the terms, or is less than this by an even number, and the number of negative zeros of P(x) is the same as the number of changes in signs of the coefficients of the terms of P( x), or is less than this by an even number. Example: f(x) = x 6 + 3x 5 4x 4 6x 3 + x 2 8x + 5 Find Numbers of Positive, Negative Zeros, and Imaginary Zeros p(x) = x 6 + 4x 3 2x 2 x 1 Your Turn Examples State the possible number of positive real zeros, negative real zeros, and imaginary zeros of p(x) = x 4 x 3 + x 2 + x + 3. What are all the zeros of f(x) = x 3 3x 2 2x + 4? More
Words: Let a and b be real numbers, and b 0. If a + bi is a zero of a polynomial function with real coefficients, then a bi is also a zero of the function. Complex Conjugate Theorem Example: If 3 + 4i is a zero of f(x) = x 3 4x 2 + 13x + 50, then 3 4i is also a zero of then function as well. Write a polynomial function of least degree with integral coefficients, the zeros of which include 4 and 4 i. Examples What is a polynomial function of least degree with integral coefficients the zeros of which include 2 and 1 + i?
Name: Rational Zero Theorem 4.6 Class: Topic: Rational Zero Theorem Date: Main Ideas: Assignment: Solve x 2 + 4x + 7 = 0 How many negative real zeros does p(x) = x 4 7x 3 + 2x 2 6x 2 have? Review What is the least degree of a polynomial function with zeros that include 5 and 3i? Rational Zero Theorem Watch YOUR P s and Q s Words: If P(x) is a polynomial function with integral coefficients, then every rational zero of P(x) = 0 is of the form p, a rational number in simplest form, where p is a factor of the constant term and q q is a factor of the leading coefficient. Example: Let f(x) = 6x 4 + 22x 3 + 11x 2 80x 40. If 4 is a zero of f(x), then 4 is a factor of -40, 3 and 3 is a factor of 6. List all of the possible rational zeros of f(x) = 3x 4 x 3 + 4. List all of the possible rational zeros of f(x) = x 4 + 7x 3 15. Your Turn List all of the possible rational zeros of f(x) = 2x 3 + x + 6.
Application Question The volume of a rectangular solid is 1120 cubic feet. The width is 2 feet less than the height, and the length is 4 feet more than the height. Find the dimensions of the solid. Find all of the zeros of f(x) = x 4 + x 3 19x 2 + 11x + 30. All Zeros Find all of the zeros of f(x) = x 4 + 4x 3 14x 2 36x + 45.
Graphing Polynomial Functions 4.7 Name: Class: Topic: Graphing Date: Main Ideas: Assignment: Given: f(x) = x 4 + 6x 3 5x 2 52x 60 End Behaviors LC: Possible Turning Points y-int Degree: RS: LS: Things We Know DesCartes Rule of Sign Change Triple/Double/Single Roots: Touch: Cross: Possible Rational Roots P s: Q s: P s Q s : Zeros/Roots/x-ints Work for depression of polynomial to find Zeros: Sketch: Remainder/Factor Theorem to check roots:
Given: f(x) = x 5 4x 4 + 14x 3 17x + 6 End Behaviors LC: Possible Turning Points y-int Degree: RS: LS: DesCartes Rule of Sign Change Triple/Double/Single Roots: Touch: Cross: Possible Rational Roots P s: Q s: P s Q s : Zeros/Roots/x-ints Your Turn Work for depression of polynomial to find Zeros: Sketch: Remainder/Factor Theorem to check roots:
Graphing Polynomial Functions 4.7 Write the possible equation for each polynomial graph shown in factored form. (y-axis is scaled by 5 s) (y-axis is scaled by 5 s) Writing from Graph (y-axis is scaled by 10 s) (y-axis is scaled by 100 s) Sketch the graph of the equation with a double root at 2, a single root at 5, a triple root at 0 and a double root at 2. Assume the leading coefficient is negative. Write the equation of the function that describes the graph. Equation: Writing Functions