Characteristics of Polynomials Worksheet
Explore and identify the key characteristics of polynomial functions, including degree, leading coefficient, end behavior, zeros, and multiplicity, suitable for Grade 12 Algebra students.
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Characteristics of Polynomials
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Read each question carefully and provide your answer in the space provided. Show all your work for full credit.
1. The of a polynomial is the highest power of the variable in the polynomial.
2. The is the coefficient of the term with the highest degree.
3. If the degree of a polynomial is even and the leading coefficient is negative, the end behavior of the graph will be on both sides.
4. A zero with an odd multiplicity will the x-axis.
1. Which of the following is NOT a characteristic of a polynomial function?
The exponents of the variables are always positive integers.
The graph is always continuous and smooth.
The domain is all real numbers.
It can have vertical asymptotes.
2. What is the end behavior of the polynomial function f(x) = -2x^3 + 5x - 1?
As x → ∞, f(x) → ∞; As x → -∞, f(x) → ∞
As x → ∞, f(x) → -∞; As x → -∞, f(x) → -∞
As x → ∞, f(x) → -∞; As x → -∞, f(x) → ∞
As x → ∞, f(x) → ∞; As x → -∞, f(x) → -∞
1. For the polynomial function f(x) = (x + 3)^2 (x - 1)(x - 4)^3:
a) What is the degree of the polynomial?
b) What are the zeros and their multiplicities?
c) Describe the end behavior of the graph.
1. A polynomial function can have a sharp corner or cusp in its graph.
True
False
2. If a polynomial has a zero with an even multiplicity, the graph will cross the x-axis at that zero.
True
False
1. Sketch a possible graph for a polynomial function with the following characteristics:
- Degree 4
- Leading coefficient is positive
- Zeros at x = -2 (multiplicity 1), x = 0 (multiplicity 2), and x = 3 (multiplicity 1)