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Chapter 7: Polynomial Functions

Chapter 7:

Polynomial Functions7.1 Polynomial Functions

Objectives:

Evaluate polynomial functions

2)Identify general shapes of graphs of polynomial functionsPolynomial FunctionsRecall that a polynomial is a monomial or sum of monomials. The expression 3x2 3x + 1 is a polynomial in one variable since it only contains one variable, x. A polynomial of degree n in one variable x is an expression of the form a0xn + a1xn-1 + + an-1x + an where the coefficients a0, a1, a2an represent real numbers, a0 is not zero, and n represents a nonnegative integer. Examples: 3x5 + 2x4 - 5x3 + x2 + 1n=5, a0=3, a1=2, a2= -5, a3=1, a4=0, and a5=1VocabularyDegree of a polynomial in one variable is the greatest exponent of its variable. The leading coefficient is the coefficient of the term with the highest degree.

PolynomialExpressionDegreeLeading CoefficientConstant909Linearx - 2 11Quadratic3x2 + 4x - 523Cubic4x3 - 634Generala0xn + a1xn-1 + + an-1x + an na0Example 1State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. 7x4 + 5x2 + x 9

8x2 + 3xy 2y2

7x6 4x3 +

x2 + 2x3 x5

Polynomial FunctionA polynomial equation used to represent a function.Examples:f(x) = 4x2 5x + 2 is a quadratic polynomial function.f(x) = 2x3 + 4x2 5x + 7 is a cubic polynomial function.

If you know an element in the domain of any polynomial function, you can find the corresponding value in the range.

Recall that f(3) can be found by evaluating the function for x = 3.

Example 2a) Show that the polynomial function f(r)=3r2 3r + 1 gives the total number of hexagons when r = 1, 2, and 3.

Find the total number of hexagons in a honeycomb with 12 rings.

Example 3Functional Values of VariablesFind p(a2) if p(x) = x3 + 4x2 5x

Find q(a +1) 2q(a) if q(x) = x2 + 3x + 4

Graphs of Polynomial FunctionsThese graphs show the MAXIMUM number of times the graph of each type of polynomial may intersect the x-axis. The x-coordinate of the point at which the graph intersects the x-axis is called a zero of a function. How does the degree compare to the maximum number of real zeros?

Notice the shapes of the graphs for even-degree polynomial functions and odd-degree polynomial functions. The degree and leading coefficient of a polynomial function determine the graphs end behavior.

Partner Practice Text p. 350 #s 1-11 allHomeworkText p. 350-351 #s 16-38 evenEnd BehaviorIs the behavior of a graph as x approaches positive infinity (+ ) or negative infinity ( ).This is represented as x + and x -

Note on x-intercepts:Graph of an even-degree function:may or may not intersect the x-axisif it intersects the x-axis in two places, the function has two real zeros. if it does not intersect the x-axis the roots of the related equation are imaginary and cannot be determined from the graphIf the graph is tangent to the x-axis, as shown above, there are two zeros that are the same number

Note on x-intercepts (ctd):Graph of an odd-degree function:ALWAYS intersect the x-axis at least onceThus, ALWAYS has at least one real zero.

Example 4a1) describe end behavior 2) determine whether it represents an odd-degree or even-degree polynomial function 3) state the number of real zeros

Example 4b1) describe end behavior 2) determine whether it represents an odd-degree or even-degree polynomial function 3) state the number of real zeros

Example 4c1) describe end behavior 2) determine whether it represents an odd-degree or even-degree polynomial function 3) state the number of real zeros

HomeworkText p. 350 #s 12-15 all Text p. 351 #s 39-48 all

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