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College Algebra Exam Review 311

College Algebra Exam Review 311 - det.V V t D det 2 4 3 P...

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7.2. SOLVING THE CUBIC EQUATION 321 Exercises 7.2 7.2.1. Show that a cubic polynomial in KOExŁ either has a root in K or is irreducible over K . 7.2.2. Verify the reduction of the monic cubic polynomial to a polynomial with no quadratic term. 7.2.3. How can you deal with a nonmonic cubic polynomial? 7.2.4. Verify in detail the derivation of Cardano’s formulas. 7.2.5. Consider the polynomial p.x/ D x 3 C 2x 2 C 2x 3 . Show that p is irreducible over Q . Hint : Show that if p has a rational root, then it must have an integer root, and the integer root must be a divisor of 3 . Carry out the reduction of p to a polynomial f without quadratic term. Use the method described in this section to find the roots of f . Use this information to find the roots of p . 7.2.6. Repeat the previous exercise with various cubic polynomials of your choice. 7.2.7. Let V denote the Vandermonde matrix 2 4 1 1 1 ˛ 1 ˛ 2 ˛ 3 ˛ 2 1 ˛ 2 2 ˛ 2 3 3 5 : (a) Show that
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Unformatted text preview: det .V V t / D det 2 4 3 P ˛ 2 i P ˛ 2 i P ˛ 3 i P ˛ 2 i P ˛ 3 i P ˛ 4 i 3 5 . (b) Use equations ( 7.2.1 ) as well as the fact that the ˛ i are roots of f.x/ D to compute that P ˛ 2 i D ± 2p , P i a 3 i D ± 3q , and P i a 4 i D 2p 2 . (c) Compute that ı 2 D ± 4p 3 ± 27q 2 . 7.2.8. (a) Show that x 3 ± 2 is irreducible in Q ŒxŁ . Compute its roots and also ı 2 and ı . (b) Q . 3 p 2/ ² R , so is not equal to the splitting field E of x 3 ± 2 . The splitting field is Q . 3 p 2;!/ , where ! D e 2±i=3 . Show that ! satisfies a quadratic polynomial over Q . (c) Show that x 3 ± 3x C 1 is irreducible in Q ŒxŁ . Compute its roots, and also ı 2 and ı . Hint: The quantity A is a root of unity, and the roots are twice the real part of certain roots of unity....
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