Exploring Graphs and Properties of Polynomial Functions Assignment

Assignment Task

1.  Create four different polynomials, in the factored form, satisfying the following conditions:

a. Polynomial 1 - non-monic, degree 4 with linear factors.

b. Polynomial 2 - degree 5 with two linear factors of multiplicity 2 and negative leading coefficient

c. Polynomial 3 - monic, degree 6 with one linear factor of multiplicity 2 and one linear factor of multiplicity

d. Polynomial 4 - monic, degree 7 with two linear factors of multiplicity 3

2. Using a graphing software, sketch the graph of each polynomial that you have chosen/created above. 

• The polynomial functions must be included and the zeros of each graph should be clearly marked. 
• Take a screenshot of each graph and attach all the graphs on your answer booklet.
• Each graph should cover at least one-third of an A4 page.

3. What are your observations about:

a. The number of zeros and the degree of a polynomial?

b. The number of turning points and the degree of a polynomial?

4. What is the significance of the leading term of a polynomial? 

Give two features which impacts the graph and explain with a help of a simple cubic polynomial.

René Descartes, a French mathematician, discovered among other things, a relationship between the number of zeros and the sign changes in the sequence of the coefficients of a polynomial.

5. What is Descartes’ Rule of signs?

Do some research and write a paragraph, about 300 words in length, about your findings. You must include a comprehensive example of how the rule is applied in relation to the zeros of a polynomial.

6. Use the software Wolfram Alpha or anything similar, to write each of the polynomials in Question1 in the expanded form.

7. By analysing each of the graphs of the polynomials that you have sketched in

Question 2, show that 8 they satisfy Descartes’ Rule of signs about the zeros of a polynomial. You will need to present full working out with clear explanation for each of the polynomial.

2. INVESTIGATION OF THE BEHAVIOUR OF A GRAPH AT THE ZEROS 

8. Use a graphing software, sketch the graphs of y = P(x) and y* = P(x) onthe same set ofaxes for 4 each of the four polynomials from Question 1.

Take a clear screenshot of each graph, with labels and place it to a word document so it can be submitted in the appropriate place on Google Classroom.

Each graph should cover at least one-third of an A4 page.

9. Now investigate the behaviour of the graphs of y = P(x) at the zeros in relation to their 6 multiplicity, the type of x-intercept and the shape of the graph in the vicinity of the zeros.

Write a paragraph of your findings with small sketches, providing suitable explanations. 

3. INEQUALITIES WITH UNKNOWN DENOMINATORS

10. You are to prepare a lesson plan on how to solve inequalities with ‘unknown’. denominators in the form ae with variations, where f(x) and g(x) are linear functions.

Your lesson plan should include the following:

a. Three different methods to solve an inequality in the form f(x)/g(x)~ <1>

f(x) and g(x) are linear functions of your choice, which should ensure that there are solutions to your inequality.

b. f(x) Two different methods to solve an inequality in the form f(x)/g(x) > 1

f(x) and g(x) are linear functions of your choice, which should ensure that there are solutions to your inequality.

Your methods in each part must be clearly explained step-by-step with justification, demonstrating your comprehensive understanding of the concepts.

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