Sunday, June 16, 2024

Algebra 2 Chapter 6 Polynomials And Polynomial Functions

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How To: Given A Polynomial Function Sketch The Graph

Algebra 2 Introduction, Basic Review, Factoring, Slope, Absolute Value, Linear, Quadratic Equations
  • Find the intercepts.
  • Check for symmetry. If the function is an even function, its graph is symmetric with respect to the y-axis, that is, f = f.If a function is an odd function, its graph is symmetric with respect to the origin, that is, f = f.
  • Use the multiplicities of the zeros to determine the behavior of the polynomial at the x-intercepts.
  • Determine the end behavior by examining the leading term.
  • Use the end behavior and the behavior at the intercepts to sketch the graph.
  • Ensure that the number of turning points does not exceed one less than the degree of the polynomial.
  • Optionally, use technology to check the graph.
  • The Intermediate Value Theorem

    In some situations, we may know two points on a graph but not the zeros. If those two points are on opposite sides of the x-axis, we can confirm that there is a zero between them. Consider a polynomial function f whose graph is smooth and continuous. The Intermediate Value Theorem states that for two numbers a and b in the domain of f, if a < b and f\left\ne f\left, then the function f takes on every value between f\left and f\left.

    We can apply this theorem to a special case that is useful for graphing polynomial functions. If a point on the graph of a continuous function f at x=a lies above the x-axis and another point at x=b lies below the x-axis, there must exist a third point between x=a and x=b where the graph crosses the x-axis. Call this point \left\right). This means that we are assured there is a value c where f\left=0.

    In other words, the Intermediate Value Theorem tells us that when a polynomial function changes from a negative value to a positive value, the function must cross the x-axis. The figure below shows that there is a zero between a and b.

    The Intermediate Value Theorem can be used to show there exists a zero.

    A General Note: Factored Form Of Polynomials

    If a polynomial of lowest degree p has zeros at x=_,_,\dots ,_, then the polynomial can be written in the factored form: f\left=a_\right)}^_}_\right)}^_}\cdots _\right)}^_} where the powers _ on each factor can be determined by the behavior of the graph at the corresponding intercept, and the stretch factor a can be determined given a value of the function other than the x-intercept.

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    Chapter 6 Polynomials And Polynomial Functions

    1 Chapter 6 Polynomials and Polynomial Functions

    2 Lesson 6-1 Polynomial Functions

    3 Polynomials A polynomial is a monomial or the sum of monomials. P a x a x… a x a n n1 n n P x 5x x 5 The exponent of the variable in a term determines the degree of the term. A polynomial shown in descending order by degree is in standard form.

    4 Polynomials Cubic Term Quadratic Term Linear Term Constant P x 3 5x x 5 Leading Coefficient

    5 Degree of a Polynomial Degree Name Using Degree Example Number of Term Name Using Number of Terms 0 Constant 1 Monomial 6 x 3 1 Linear Binomial 3x Quadratic Binomial 3 Cubic 3 3 Trinomial x 5x x 4 Quartic 4 Binomial x 3x 5 Quintic x 5 3x x 4 4 Polynomial of 4 terms

    6 Example 1 Page 303, # Write each polynomial in standard form. Then classify it by degree and by number terms. 5 3x 3x 5 linear terms binomial

    7 Example 1 Page 303, #1 Write each polynomial in standard form. Then classify it by degree and by number terms. x x x 4 x 4 3x quartic terms binomial

    8 Example Page 304, #34 Simplify. Classify each result by number of terms. 8d 3 7 d d 7 d 6 3 9d 13 terms Binomial

    9 Example Page 304, #4 Simplify. Classify each result by number of terms. 1x 3 5x 3 4x x 3 3 1x 5x 3 4x 319x 4 3 4x 4 3x 3 5x 54 4 terms Polynomial of 4 terms

    10 Example Page 304, #46 Find each product. Classify the result by number of terms. 4 1 x x x x 4x 1 8x x 3 terms binomial

    12 Lesson 6-, Part 1 Polynomial and Linear Functions

    16 Lesson 6-, Part Polynomial and Linear Functions

    Presentation On Theme: Algebra 2 Chapter 6 Notes Polynomials And Polynomial Functions Algebra 2 Chapter 6 Notes Polynomials And Polynomial Functions Presentation Transcript:

    1 Algebra 2 Chapter 6 Notes Polynomials and Polynomial Functions Algebra 2 Chapter 6 Notes Polynomials and Polynomial Functions

    2 NOTES: Page 72, Section 6.1 Using Properties of Exponents Product of Powers Property a m a n = a m + n 5 2 5 3 = 5 2 + 3 = 5 5 Power of a Power Property n = a m n 3 = 5 6 Power of a Product Property m = a m b m 2 = 5 2 4 2 Negative Exponent Property a m = 1 a m 5 2 = 1 5 2 Zero Exponent Property a 0 = 15 0 = 1 Quotient of Powers Property a m = a m n a n 5 3 = 5 3 2 = 5 1 5 2 Power of a Quotient Property a m = a m b b m 5 3 = 5 3 4 4 3, a 0

    3 NOTES: Page 72a, Section 6.1 Using Properties of Exponents Evaluating Numerical Expressions 4 = 2 3 4 = 2 12 = 4096 3 2 4 = 3 2 4 2 = 9 16 6 4 = 6 + 4 = 2 = 1 2 = 1 25 Simplifying Algebraic Expressions r 2 s 5 = 2 2 = r 2 s 10 = r 2 s 10 2 b 5 b= 7 2 b 6 b 5 b= 49 b 6 + 5 + 1 = 49 b 0 = 49 2 x 3 y 1 = x 2 y 4 x 3 y 1 = x 23 y 4 = x 1 y 5 = y 5 x 1

    4 NOTES: Page 73, Section 6.2 Evaluating and Graphing Polynomial Functions f = a n x n + a n 1 x n 1 + + a 1 x + a 0 Leading coefficient Constant term Degree of polynomial Polynomial Function in standard form: Descending order of exponents from left to right. DegreeTypeStandard form 0Constantf = a 0 1Linearf = a 1 x + a 0 2Quadraticf = a 2 x 2 + a 1 x + a 0 3Cubicf = a 3 x 3 + a 2 x 2 + a 1 x + a 0 4Quarticf = a 4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0

    12 NOTES: Page 75a, Section 6.3 Multiplying Polynomials

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    How To: Given A Graph Of A Polynomial Function Write A Formula For The Function

  • Identify the x-intercepts of the graph to find the factors of the polynomial.
  • Examine the behavior of the graph at the x-intercepts to determine the multiplicity of each factor.
  • Find the polynomial of least degree containing all of the factors found in the previous step.
  • Use any other point on the graph to determine the stretch factor.
  • Chapter 6 Polynomial Functions Algebra 2 Warm Up

    Chapter 6 Polynomial Functions Algebra 2

    Warm Up

    6 -1 Polynomial Functions A monomial is an expression that is either a real number, a variable or a product of real numbers and variables. A polynomial is a monomial or the sum of monomials. The exponent of the variable in a term determines the degree of that term. Standard form of a polynomial has the variable in descending order by degree.

    6 -1 Polynomial Functions

    6 -1 Polynomial Functions The degree of a polynomial is the greatest degree of any term in the polynomial

    6 -1 Polynomial Functions Write each polynomial in standard form and classify it by degree.

    6 -2 Polynomials and Linear Factors You can write a polynomial as a product of its linear factors

    6 -2 Polynomials and Linear Factors You can sometimes use the GCF to help factor a polynomial. The GCF will contain variables common to all terms, as well as numbers

    6 -2 Polynomials and Linear Factors

    6 -2 Polynomials and Linear Factors

    6 -2 Polynomials and Linear Factors If a linear factor of a polynomial is repeated, the zero is repeated. A repeated zero is called a multiple zero. A multiple zero has multiplicity equal to the number of times the zero occurs.

    6 -2 Polynomials and Linear Factors

    6 -2 Polynomials and Linear Factors page 323 odd you do NOT need to graph the functions.

    warm up

    6 -3 Dividing Polynomials

    Synthetic Division

    6 -4 Solving polynomial equations

    Solve for all three roots

    solving using a quadratic model

    homework p 354 odd

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    Example: Identifying Zeros And Their Multiplicities

    Use the graph of the function of degree 6 to identify the zeros of the function and their possible multiplicities.

    The polynomial function is of degree n which is 6. The sum of the multiplicities must be 6.

    Starting from the left, the first zero occurs at x=-3. The graph touches the x-axis, so the multiplicity of the zero must be even. The zero of 3 has multiplicity 2.

    The next zero occurs at x=-1. The graph looks almost linear at this point. This is a single zero of multiplicity 1.

    The last zero occurs at x=4. The graph crosses the x-axis, so the multiplicity of the zero must be odd. We know that the multiplicity is 3 and that the sum of the multiplicities must be 6.

    Presentation On Theme: Chapter 6

    Solving Polynomial Equations By Factoring and Using Synthetic Division – Algebra 2 & Precalculus

    1 Chapter 6 – Polynomial FunctionsAlgebra 2

    2 6.3 – Dividing Polynomials6.4 – Factoring Polynomials6.5 – Finding Real Roots of Polynomial Equation

    3 6.1 – PolynomialsAlgebra 2

    4 6.1Algebra 2 A monomial is a number or a product of numbers and variables with whole number exponents. Ex: 2xA polynomial is multiple monomials or terms put together in an expressionex: 3×2 + 2x + 1**Polynomials have no variables in the denominators or exponents, no roots or absolute values of variables, and all variables have whole number exponents.**12a7Polynomials:3x42z12 + 9z30.15x1013t2 t385y212Not polynomials:3x|2b3 6b|m0.75 mThe degree of a monomial is the sum of the exponents of the variables.

    5 A. z6 B. 5.6 C. 8xy3 D. a2bc3 Identify the degree of each monomial. z66.1Example 1Identifying the Degree of a MonomialIdentify the degree of each monomial.A. z6B. 5.6z6Identify the exponent.5.6 = 5.6x0Identify the exponent.The degree is 6.The degree is 0.C. 8xy3D. a2bc38x1y3Add the exponents.a2b1c3Add the exponents.The degree is 4.The degree is 6.

    6 6.1An degree of a polynomial is given by the term with the greatest degree or powerStandard Form: When the expression is written by using highest powers in descending orderWhen in standard form with the highest degree first, the leading coefficient is the number in front of that highest power term

    8 Math Joke Knock, Knock Whos there? Polly Polly who?6.1Math JokeKnock, KnockWhos there?PollyPolly who?Polynomial, Why the third degree?

    34 6.3Just Read

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    Chapter : Polynomials And Polynomial Functions

    6.1 Using Properties of ExponentsGoals: How to use properties of exponents to evaluate and simplify expressions involving powers and to use exponents and scientific notation to solve real-life problems.6.2 Evaluating and Graphing Polynomial FunctionsGoals: How to evaluate and graph a polynomial function.6.3 Adding, Subtracting, and Multiplying PolynomialsGoals: Add, subtract, and multiply polynomials.6.4 Factoring and Solving Polynomial EquationsGoals: How to factor polynomial expressions and use factoring to solve polynomial equations.6.5 The Remainder and Factor TheoremsGoals: How to divide polynomials and relate the result to the remainder theorem and the factor theorem.6.4 – 6.6 Quiz 6.7 Using the Fundamental Theorem of AlgebraGoals: How to use the fundamental theorem of algebra to determine the number of zeros of a polynomial function and how to use technology to approximate the real zeros of a polynomial function.

    Example: Using The Intermediate Value Theorem

    Show that the function f\left=^-5^+3x+6 has at least two real zeros between x=1 and x=4.

    To start, evaluate f\left at the integer values x=1,2,3,\text4.

    3 2

    We see that one zero occurs at x=2. Also, since f\left is negative and f\left is positive, by the Intermediate Value Theorem, there must be at least one real zero between 3 and 4.

    We have shown that there are at least two real zeros between x=1 and x=4.

    Analysis of the Solution

    We can also graphically see that there are two real zeros between x=1 and x=4.

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    Example: Writing A Formula For A Polynomial Function From Its Graph

    Write a formula for the polynomial function.

    This graph has three x-intercepts: x = 3, 2, and 5. The y-intercept is located at . At x = 3 and x = 5, the graph passes through the axis linearly, suggesting the corresponding factors of the polynomial will be linear. At x = 2, the graph bounces off the x-axis at the intercept suggesting the corresponding factor of the polynomial will be second degree . Together, this gives us


    To determine the stretch factor, we utilize another point on the graph. We will use the y-intercept , to solve for a.

    \beginf\left=a\left^\left\hfill \\ \text-2=a\left^\left\hfill \\ \text-2=-60a\hfill \\ \texta=\frac\hfill \end

    The graphed polynomial appears to represent the function f\left=\frac\left^\left.

    Basic Knowledge Of Polynomial Functions

    A polynomial is a mathematical expression constructed with constants and variables using the four operations:

    5x4+4x3+3x2+2 x+1 4

    In other words, we have been calculating with various polynomials all along. When two polynomials are divided it is called a rational expression.

    In such cases you must be careful that the denominator does not equal zero. Division by zero is not defined and thus x may not have a value that allows the denominator to become zero. Otherwise, any other value may be substituted for x.



    x must not have the value of 6 since 6-6=0.

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    Writing Formulas For Polynomial Functions

    Now that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. Because a polynomial function written in factored form will have an x-intercept where each factor is equal to zero, we can form a function that will pass through a set of x-intercepts by introducing a corresponding set of factors.

    A General Note: Graphical Behavior Of Polynomials At X

    If a polynomial contains a factor of the form ^, the behavior near the x-intercept h is determined by the power p. We say that x=h is a zero of multiplicityp.

    The graph of a polynomial function will touch the x-axis at zeros with even multiplicities. The graph will cross the x-axis at zeros with odd multiplicities.

    The sum of the multiplicities is the degree of the polynomial function.

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    Example: Using Local Extrema To Solve Applications

    An open-top box is to be constructed by cutting out squares from each corner of a 14 cm by 20 cm sheet of plastic then folding up the sides. Find the size of squares that should be cut out to maximize the volume enclosed by the box.

    We will start this problem by drawing a picture like the one below, labeling the width of the cut-out squares with a variable, w.

    Notice that after a square is cut out from each end, it leaves a \left cm by \left cm rectangle for the base of the box, and the box will be w cm tall. This gives the volume

    \beginV\left=\left\leftw\hfill \\ \textV\left=280w – 68^+4^\hfill \end

    Notice, since the factors are w, 20 – 2w and 14 – 2w, the three zeros are 10, 7, and 0, respectively. Because a height of 0 cm is not reasonable, we consider only the zeros 10 and 7. The shortest side is 14 and we are cutting off two squares, so values w may take on are greater than zero or less than 7. This means we will restrict the domain of this function to 0< w< 7. Using technology to sketch the graph of V\left on this reasonable domain, we get a graph like the one above. We can use this graph to estimate the maximum value for the volume, restricted to values for w that are reasonable for this problem, values from 0 to 7.

    From this zoomed-in view, we can refine our estimate for the maximum volume to about 339 cubic cm which occurs when the squares measure approximately 2.7 cm on each side.

    Example: Sketching The Graph Of A Polynomial Function

    Introduction to Polynomial Functions (Precalculus – College Algebra 27)

    Sketch a possible graph for f\left=-2^\left.

    This graph has two x-intercepts. At x = 3, the factor is squared, indicating a multiplicity of 2. The graph will bounce off the x-intercept at this value. At x = 5, the function has a multiplicity of one, indicating the graph will cross through the axis at this intercept.

    The y-intercept is found by evaluating f.

    \begin\hfill \\ f\left=-2^\left\hfill \\ \textf\left=-2\cdot 9\cdot \left\hfill \\ \textf\left=90\hfill \end

    The y-intercept is .

    Additionally, we can see the leading term, if this polynomial were multiplied out, would be -2^, so the end behavior, as seen in the following graph, is that of a vertically reflected cubic with the outputs decreasing as the inputs approach infinity and the outputs increasing as the inputs approach negative infinity.

    To sketch the graph, we consider the following:

    • As x\to -\infty the function f\left\to \infty , so we know the graph starts in the second quadrant and is decreasing toward the x-axis.
    • Since f\left=-2^\left is not equal to f, the graph does not have any symmetry.
    • At \left the graph bounces off of the x-axis, so the function must start increasing after this point.
    • At , the graph crosses the y-axis.

    Somewhere after this point, the graph must turn back down or start decreasing toward the horizontal axis because the graph passes through the next intercept at .

    The complete graph of the polynomial function f\left=-2^\left is as follows:

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