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Precalculus: Polynomial Functions

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  • What defines a polynomial function?
    A polynomial function is defined as \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\) where n is a nonnegative integer and a_n \(\neq\) 0.
  • What is the degree of a polynomial?
    The degree of a polynomial is the highest power of the variable with a nonzero coefficient.
  • Is \(f(x) = x + 3\) a polynomial? If yes, what is its degree?
    Yes, it is a polynomial of degree 1.
  • Is \(g(x) = 7x^5 - \pi x^3 + \frac{1}{5}x\) a polynomial? If yes, what is its degree?
    Yes, it is a polynomial of degree 5.
  • Is \(h(x) = 8x^3 - x^2 + \frac{2}{x}\) a polynomial? Why or why not?
    No, because it contains a term with x in the denominator, which is not allowed in polynomials.
  • What is the end behavior of a polynomial with odd degree and positive leading coefficient?
    The graph falls to the left and rises to the right.
  • What is the end behavior of a polynomial with odd degree and negative leading coefficient?
    The graph rises to the left and falls to the right.
  • What is the end behavior of a polynomial with even degree and positive leading coefficient?
    The graph rises to the left and rises to the right.
  • What is the end behavior of a polynomial with even degree and negative leading coefficient?
    The graph falls to the left and falls to the right.
  • How do you find the roots of a polynomial?
    Set the polynomial equal to zero and solve by factoring or using the quadratic formula.
  • What does the multiplicity of a root tell you about the graph at that root?
    If the root has even multiplicity, the graph touches and turns around at the x-axis. If odd multiplicity, the graph crosses the x-axis.
  • What is the maximum number of turning points for a polynomial of degree n?
    At most \(n - 1\) turning points.
  • How does the graph behave near a root with multiplicity greater than 1?
    The graph flattens out near the root.
  • What is the y-intercept of a polynomial function?
    The y-intercept is the value of the polynomial at \(x=0\), i.e., \(f(0)\).
  • How can symmetry help in graphing polynomials?
    If \(f(-x) = f(x)\), the graph is symmetric about the y-axis. If \(f(-x) = -f(x)\), the graph is symmetric about the origin.
  • Is the function \(f(x) = x^2\) even, odd, or neither?
    It is even because \(f(-x) = f(x)\).
  • Is the function \(f(x) = x^3 + 4x^2\) even, odd, or neither?
    It is neither even nor odd.
  • What happens to the graph at a root with multiplicity 2?
    The graph touches the x-axis and bounces off (turns around) at that root.
  • What happens to the graph at a root with multiplicity 3?
    The graph crosses the x-axis and flattens near that root.
  • How do you determine if a function is a polynomial from its formula?
    Check if all exponents of x are whole numbers and no variables appear in denominators or under roots.