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