1.5A Polynomial functions and complex zeros
What is a zero of a polynomial function?
Define the term 'multiplicity' in the context of polynomial function zeros.
What determines the number of complex zeros a polynomial function has?
What does it mean for a polynomial function's graph to have an x-intercept at a point (a,\, 0)?
How does the presence of a non-real zero affect the other zeros of a polynomial function?
Determine the real and complex zeros of the following polynomial functions:
For the given polynomial functions, identify the zeros and their multiplicities.
Determine the zeros and their multiplicities for the following polynomial functions:
Consider the following polynomial function: f(x) = x^3 - 9x^2 + 26x - 24
Identify the x-intercepts of the function and their corresponding points (a,\, 0) on the graph.
Write the polynomial function in factored form based on the x-intercepts found in part a.
For each of the following polynomial functions:
Determine the multiplicity of each zero.
Describe the behavior of the graph near each zero.
The graph of a polynomial function is shown.
Identify the multiplicity of each zero.
Describe the behavior of the graph near each zero.
A polynomial function is represented by the following table of values. Calculate the successive differences and determine the degree of the polynomial function:
| x | -5 | -3 | -1 | 1 | 3 | 5 |
|---|---|---|---|---|---|---|
| y | 9 | 5 | 1 | 1 | 5 | 9 |
The polynomial function k(x) = x^3 - 3x^2 - 4x + 12 has a complex zero 2 + i. Write the polynomial in factored form using its real and non-real zeros.
Consider the polynomial function m(x) = x^5 - 7x^3 + 6x. What is the significance of the polynomial having a zero with even multiplicity? How does this affect the graph of the function?
A polynomial function has a degree of 3 and has zeros at x = -1, x = 2, and x = 4. The zero at x = -1 has a multiplicity of 2.
Write the polynomial function in standard form.
Determine the regions of the domain where the function is positive, negative, and zero.
Consider a polynomial function q(x) with the following properties:
Has a leading coefficient of 1
Has zeros at x = -2, x = 1 + i, and x = 1 - i
Write the polynomial function q(x) in factored form.
Determine the real and complex zeros of the function and their multiplicities.
Identify any connections between the complex zeros and their conjugates, using the complex conjugate zeros theorem.