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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. 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.

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 & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.3: Zeros Of Polynomials
Problem 41E
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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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