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Prove that if a real number c satisfies a polynomial
Chapter 4, Problem 31E(choose chapter or problem)
Prove that if a real number c satisfies a polynomial equation of the form
\(r_{3} x^{3}+r_{2} x^{2}+r_{1} x+r_{0}=0\),
where \(r_{0}, r_{1}, r_{2}, \text { and } r_{3}\) are rational numbers, then c satisfies an equation of the form
\(n_{3} x^{3}+n_{2} x^{2}+n_{1} x+n_{0}=0\),
where \(n_{0}, n_{1}, n_{2}, \text { and } n_{3}\) are integers.
Text Transcription:
r_3x^3 + r_2x^2 + r_1x + r_0 = 0
r_0, r_1, r_2, and r_3
n_3x^3 + n_2x^2 + n_1x + n_0 = 0
n_0, n_1, n_2, and n_3
Questions & Answers
QUESTION:
Prove that if a real number c satisfies a polynomial equation of the form
\(r_{3} x^{3}+r_{2} x^{2}+r_{1} x+r_{0}=0\),
where \(r_{0}, r_{1}, r_{2}, \text { and } r_{3}\) are rational numbers, then c satisfies an equation of the form
\(n_{3} x^{3}+n_{2} x^{2}+n_{1} x+n_{0}=0\),
where \(n_{0}, n_{1}, n_{2}, \text { and } n_{3}\) are integers.
Text Transcription:
r_3x^3 + r_2x^2 + r_1x + r_0 = 0
r_0, r_1, r_2, and r_3
n_3x^3 + n_2x^2 + n_1x + n_0 = 0
n_0, n_1, n_2, and n_3
ANSWER:Solution:Step 1:The objective of this question is to prove that if a real number c satisfies a polynomial equation of the form.r3 x3 + r2 x2 + r1 x + r0 = 0,