53-56 - Polynomials with Specified Zeros Find a polynomial with real coefficients of the specified degree that satisfies the given conditions. 53. Degree 3; zeros -, 2, 3; constant coefficient 12 54. Degree 4; zeros 4 (multiplicity 2) and 3i; integer coefficients; coefficient of x² is -25 55. Complex Zeros of Polynomials Does there exist a polyno- mial of degree 4 with integer coefficients that has zeros i, 2i, 3i, and 4i? If so, find it. If not, explain why. 56. Polynomial with no Real Roots Prove that the equation 3x* + 5x² + 2 = 0 has no real root.

Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter3: Polynomial And Rational Functions
Section3.CR: Chapter Review
Problem 55E
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53–56 ■ Polynomials with Specified Zeros Find a polynomial
with real coefficients of the specified degree that satisfies the
given conditions.

53-56 - Polynomials with Specified Zeros Find a polynomial
with real coefficients of the specified degree that satisfies the
given conditions.
53. Degree 3; zeros -, 2, 3; constant coefficient 12
54. Degree 4; zeros 4 (multiplicity 2) and 3i; integer
coefficients; coefficient of x² is -25
55. Complex Zeros of Polynomials Does there exist a polyno-
mial of degree 4 with integer coefficients that has zeros i, 2i,
3i, and 4i? If so, find it. If not, explain why.
56. Polynomial with no Real Roots Prove that the equation
3x* + 5x² + 2 = 0 has no real root.
Transcribed Image Text:53-56 - Polynomials with Specified Zeros Find a polynomial with real coefficients of the specified degree that satisfies the given conditions. 53. Degree 3; zeros -, 2, 3; constant coefficient 12 54. Degree 4; zeros 4 (multiplicity 2) and 3i; integer coefficients; coefficient of x² is -25 55. Complex Zeros of Polynomials Does there exist a polyno- mial of degree 4 with integer coefficients that has zeros i, 2i, 3i, and 4i? If so, find it. If not, explain why. 56. Polynomial with no Real Roots Prove that the equation 3x* + 5x² + 2 = 0 has no real root.
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