Curriculum placement
Precalculus. This describes the skill, not a learner estimate.
Fundamental theorem of algebra
Explain polynomial root existence and verify the quadratic case
Problem types
- Count the roots of a polynomial
Builds on
Leads to
- End behavior of a polynomial · See it applied
A degree-n polynomial has n complex zeros counting multiplicity, so its graph crosses or touches the x-axis at most n times, and its nonreal zeros come in conjugate pairs that never appear as x-intercepts.
- Asymptotes, holes, and intercepts · See it applied
The number of zeros of the numerator and denominator gives the x-intercepts and vertical asymptotes of a rational function.
- Zeros and factors of a polynomial · Extend the idea
A polynomial of degree n has n complex zeros counted with multiplicity, which is used to count and find zeros.
Related skills
- How many times a line meets a curve · Related idea
Counting intersections of a curve and a line counts roots of a polynomial, which the fundamental theorem bounds by its degree.
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