Question

show that there exists no rational number x such that x^2 = 3

Answer #1

Prove, that between any rational numbers there exists
an irrational number.

Prove the following: (By contradiction)
If p,q are rational numbers, with p<q, then there exists a
rational number x with p<x<q.

Irrational Numbers
(a) Prove that for every rational number µ > 0, there exists
an irrational number λ > 0 satisfying λ < µ.
(b) Prove that between every two distinct rational numbers there
is at least one irrational number. (Hint: You may find (a)
useful)

f(x) = 3x^4-4x^3-3x^2-x-2
find the rational roots by the rational root test

Prove that following equations have no rational solutions.
a) x^3 + x^2 = 1
b) x^3 + x + 1 = 0
d) x^5 + x^4 + x^3 + x^2 + 1 = 0

Given the rational function h(x) = (x^3 +5x^2-2x-24)/(x^2-9),
find the following information
a. Use your knowledge of polynomials to factor the numerator and
the denominator of h(x) to rewrite the function in a different
form.
b. Use long division to find the quotient and the remainder to
rewrite h(x) in a different form.
c. What is the woman of h?
d. Identify the vertical asymptotes.
e. Determine the exact location of any removable discontinuity
if one exists.

graph the following rational function y=(3x-3)/x-2

(1) Let x be a rational number and y be an irrational. Prove
that 2(y-x) is irrational
a) Briefly explain which proof method may be most appropriate to
prove this statement. For example either contradiction,
contraposition or direct proof
b) State how to start the proof and then complete the proof

If
a<x<b and a,b are any numbers. Find a rational number between
a and b such that x does not equal the rational number.

Prove by contradiction that 5√ 2 is an irrational number. (Hint:
Dividing a rational number by another rational number yields a
rational number.)

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