Do Now: Solve the inequalityDo Now: Solve the inequality352 2 xx

Academy Algebra II/TrigAcademy Algebra II/Trig

5.1: Polynomial Functions and Models5.1: Polynomial Functions and Models

HW: p.340 (12, 13, 17-20, 40, 41, 43, HW: p.340 (12, 13, 17-20, 40, 41, 43,

45-47 – parts a,d,e only)45-47 – parts a,d,e only)

Test 4.3-4.5, 5.1, 5.5-5.6:Test 4.3-4.5, 5.1, 5.5-5.6:

VocabularyVocabulary

• Polynomial Function = a function in the form: where , exponents are whole #’s, and coefficients are real.

• Standard Form = terms are written in descending order of exponents.

• Degree = the highest exponent.

011

1 ...)( axaxaxaxf nn

nn

0na

Common Polynomial FunctionsCommon Polynomial FunctionsDegree Type Example

0 constant f(x)=14

1 linear f(x)=5x – 7

2 quadratic f(x)=x2+x-9

3 cubic f(x)=x3-x2

4 quartic f(x)=x4+x-1

Decide whether the function is a Decide whether the function is a polynomial function. If so, write it in polynomial function. If so, write it in

standard form and state its degree, type, standard form and state its degree, type, and leading coefficient.and leading coefficient.

2

24

37g(x) .)2

34

1)( .)1

πxx

xxxh

5

12

6.02)( .)4

35)( .)3

xxxk

xxxxf

x

Identify the real zeros of a polynomial function and their multiplicity.

• If a polynomial is factored completely, it is easy to solve the question f(x) = 0 using the zero-product property.

Example: Find the real zeros of the function: 31)( 2 xxxf

Real Zeros• For the polynomial:

• 7 is a zero of multiplicity 1 because the exponent on the factor of x – 7 is 1, • -3 is a zero of multiplicity 2 because the exponent on the factor of x + 3 is 2.

4

2

2

1325)(

xxxxf

List each real zero and its multiplicity.

2373)( xxxf

Form a polynomial whose real zeros and degree are given.

1.) Zeros: -3, 0, 4; degree 3

2.) Zeros: -1, multiplicity 1; 3, multiplicity 2; degree 3

Graphs of a polynomial function• Graphs are smooth (no corners) and

continuous (no breaks). – Determine which graphs are not polynomials.

Graphs of a polynomial function – turning points.

• If f is a polynomial function of degree n, then f has at most n – 1 turning points.

End Behaviors of a End Behaviors of a Polynomial FunctionPolynomial Function

• Degree: Even• Leading Coefficient:

Positive

• Degree: Even• Leading Coefficient:

Negative

End Behaviors of a End Behaviors of a Polynomial FunctionPolynomial Function

• Degree: Odd• Leading Coefficient:

Positive

• Degree: Odd• Leading Coefficient:

Negative

Do Now: Which of the graphs could be f(x) = x4 + 5x3 + 5x2 – 5x – 3?

• Hint: Identify the y-intercept to help eliminate options.

Academy Algebra II/TrigAcademy Algebra II/Trig

5.1: Finish5.1: Finish

HW: p.341-342 (57-60 all; 65,70,74 HW: p.341-342 (57-60 all; 65,70,74 – part f: like class work)– part f: like class work)

Graph the polynomial. Label intercepts, Graph the polynomial. Label intercepts, determine turning points, and end determine turning points, and end

behavior. (May use graphing calculator for behavior. (May use graphing calculator for shape between intercepts.)shape between intercepts.) 21)( xxf

Graph the polynomial. Label intercepts, Graph the polynomial. Label intercepts, determine turning points, and end determine turning points, and end

behavior. (May use graphing calculator for behavior. (May use graphing calculator for shape between intercepts.)shape between intercepts.)

3)( 2 xxxf

Graph the polynomial. Label intercepts, Graph the polynomial. Label intercepts, determine turning points, and end determine turning points, and end

behavior. (May use graphing calculator for behavior. (May use graphing calculator for shape between intercepts.)shape between intercepts.) 421)( xxxxf