Practice Methods For Finding Irreducible Factors (21..1)

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6 sample questions

Question 1.
What is a root of a polynomial?
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Think about what happens when you substitute a number into the polynomial.
Question 2.
Define a monic polynomial.
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What is special about the leading coefficient?
3Question 3.
What does the factor theorem state?
- If α is a root
- then (x - α) is a factor.
- All roots are irrational.
- Polynomials cannot have real coefficients.
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Think about what happens if substituting α into the polynomial results in zero.
4Question 4.
True or False: A polynomial of degree n can have at most n roots.
- True
- False
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Relate to polynomial properties.
Question 5.
Given the polynomial x^5 + x^3 + x^2, find its irreducible factors over the field of integers.
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Look for integers that satisfy the equation.
Question 6.
Using the polynomial x^4 + 2x^2 + 1, identify and justify its potential irreducibility.
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Check if your polynomial resembles the form of squares.

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