Polynomials: degree, roots, factoring
Polynomials with real coefficients: degree, sum and product, root and multiplicity, factoring by (x − a), maximum number of roots, complete factoring of a degree 2 or 3 polynomial with an obvious root.
- 26 cards
- 7 topics
- Free
The outline, topic by topic
Each topic is weighted the way the exam weighs it.
- Concepts
- Questions
- Sort and link
- The numbers
- The curves
- Scenarios
- Understand
A few cards from the deck
Degree of a polynomial: what is it?
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The largest exponent of x whose coefficient is not zero; the matching coefficient is called the leading coefficient.
deg(PQ) = deg P + deg Q, whereas deg(P + Q) can be smaller than both: x² + 1 and −x² + x have sum x + 1, of degree 1, since the x² terms cancel.
How many distinct real roots does P(x) = (x − 1)²(x + 2) have?
- 2
- 3
- 1
- 0
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2
P vanishes at 1 (double root) and at −2 (simple root): two distinct values. Counting with multiplicity would give 3; that is not what is asked.
A non-zero polynomial of degree n has at most … real roots.
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A non-zero polynomial of degree n has at most n real roots.
Each root extracts a factor of degree 1: after k distinct roots, a polynomial of degree n − k remains. A degree-3 polynomial thus has at most 3 distinct roots, and at least one, since its limits at −∞ and +∞ have opposite signs.