Complex Numbers – 4.2 and 4.3
Sections 4.2 and 4.3 – Zeros of Polynomial Functions
Objective: Sketching complete graphs of polynomial functions (if possible) using the end behavior and the real zeros.
1) For each polynomial function do the following:
a) Give the leading term and end behavior
b) Find ALL the zeros
c) Classify them as rational-real, irrational-real, imaginary, complex.
d) Identify the x-intercepts
e) Write the polynomial as a product of linear factor
f) Sketch a complete graph (if possible) of the polynomial function.
1) fx=(x2-4)(x2-5)(x-3)
2) fx=x2+4
3) fx=(x-5)(x2+9)
4) fx=x2-2(x2-2x+2)
5) fx=-0.5x2+1(x2+2x-2)
6) fx=x2+4x2-3x2-9
Sections 4.2 and 4.3 – Zeros of Polynomial Functions
Sections 4.2 and 4.3 – Zeros of Polynomial Functions
Finding all zeros of a Polynomial Function when it is given on the form
We’ll be getting some help from the calculator. Here is the method we’ll use:
2) For the polynomial function fx=3x3+8x2-7x-12.
a) How many complex zeros does the polynomial have?
b) Enter the function in the Y= of the calculator
c) How many real zeros (x-intercepts) does the function have? Find them by using the ZERO feature in the calculate menu. Classify them as rational-real, irrational-real, imaginary, complex.
d) Write the function in factored form:
Sections 4.2 and 4.3 – Zeros of Polynomial Functions
3) Given fx=x3+6x2+6x-4
a) How many complex zeros does the polynomial have?
b) Enter the function in the Y= of the calculator
c) According to the graph, how many real zeros (x-intercepts) does the function have? Find them with the ZERO feature of the calculator.
1. Are they all rational numbers? (the listing of the p/q helps you identify the rational zeros)
d) To find the irrational zeros we need to do the following:
1. Once a rational zero is identified, write the corresponding linear factor of the function.
2. Find the other factor of the function by using long division
3. Use the quadratic formula to find the zeros of any quadratic polynomial.
e) Classify them as rational-real, irrational-real, imaginary, complex.
f) Write the function in factored form:
Sections 4.2 and 4.3 – Zeros of Polynomial Functions
4) fx=3x3-7x2+12x-28
a) How many complex zeros does the polynomial have?
b) Enter the function in the Y= of the calculator
c) According to the graph, how many real zeros (x-intercepts) does the function have? Find them with the ZERO feature of the calculator.
a. Are they all rational numbers? (the listing of the p/q helps you identify the rational zeros)
d) To find the irrational/complex zeros we need to do the following:
a. Once a rational zero is identified, write the corresponding linear factor of the function.
b. Find the other factor of the function by using long division
c. Use the quadratic formula to find the zeros of any quadratic polynomial.
e) Classify them as rational-real, irrational-real, imaginary, complex.
f) Write the function in factored form:
Sections 4.2 and 4.3 – Zeros of Polynomial Functions
5) Use the Conjugate Pairs Theorem
6) Solve the following problems
Sections 4.2 and 4.3 – Zeros of Polynomial Functions
7) Solve the following problems
Sections 4.2 and 4.3 – Zeros of Polynomial Functions
8) Solve the equations:
a. x4-x3+2x2-4x-8=0
b. 2x3-3x2-3x-5=0
Sections 4.2 and 4.3 – Zeros of Polynomial Functions
Intermediate Value Theorem – due next class
9) Go to section 4.2 of the book, page 212 and copy the Intermediate Value Theorem along with the graph and explanations for figure 29. Do you understand what it means?
10) Show below the work for example 11, page 212
11) Now solve the suggested problem 87 on page 215.
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