Polynomial Functions and Graphs
What you must understand
A polynomial function is a sum of terms with nonnegative integer exponents. Its degree controls the end behavior: even degree means both ends point the same way, odd degree means they point opposite ways, with the sign of the leading coefficient determining direction. The leading coefficient test, the number of real zeros (at most the degree), and the multiplicity of each root all shape the graph. A root of even multiplicity touches the x-axis and turns around; odd multiplicity crosses. Turning points number at most degree minus one. You should be able to evaluate polynomials with the remainder and factor theorems, add, subtract, and multiply polynomials, and divide them using long division or synthetic division. Zeros correspond to factors, and the y-intercept is the constant term.
Traps to watch
- Reversing end behavior when the leading coefficient is negative.
- Confusing the number of turning points with the number of zeros.
- Sign errors in synthetic division, especially dropping a zero placeholder for a missing term.
- Assuming every zero is a crossing point when multiplicity may be even.
- Forgetting that the degree caps, but does not guarantee, the number of real roots.
Worked example
Describe f(x) = -2x^3(x - 1)^2. Degree is 5 (odd), leading coefficient negative, so as x goes to positive infinity f goes to negative infinity and as x goes to negative infinity f goes to positive infinity. There is a zero at x = 0 of multiplicity 3 (crossing) and at x = 1 of multiplicity 2 (touching).
Rule of thumb
Degree and leading coefficient set the ends; multiplicities set whether the graph crosses or bounces at each zero.