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Обучение Руководства > College Algebra

Key Concepts & Glossary

Key Concepts

  • Polynomial functions of degree 2 or more are smooth, continuous functions.
  • To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero.
  • Another way to find the x-intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the x-axis.
  • The multiplicity of a zero determines how the graph behaves at the x-intercepts.
  • The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity.
  • The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.
  • The end behavior of a polynomial function depends on the leading term.
  • The graph of a polynomial function changes direction at its turning points.
  • A polynomial function of degree n has at most n – 1 turning points.
  • To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most n – 1 turning points.
  • Graphing a polynomial function helps to estimate local and global extremas.
  • The Intermediate Value Theorem tells us that if f(a)andf(b)f\left(a\right) \text{and} f\left(b\right) have opposite signs, then there exists at least one value c between a and b for which f(c)=0f\left(c\right)=0.

Glossary

global maximum
highest turning point on a graph; f(a)f\left(a\right) where f(a)f(x)f\left(a\right)\ge f\left(x\right) for all x.
global minimum
lowest turning point on a graph; f(a)f\left(a\right) where f(a)f(x)f\left(a\right)\le f\left(x\right) for all x.
Intermediate Value Theorem
for two numbers a and b in the domain of f, if a<ba<b and f(a)f(b)f\left(a\right)\ne f\left(b\right), then the function f takes on every value between f(a)f\left(a\right) and f(b)f\left(b\right); specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the x-axis
multiplicity
the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form (xh)p{\left(x-h\right)}^{p}, x=hx=h is a zero of multiplicity p.
 

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