...(x — a)2 + b2, and hence if a + ib is a root of /(ж), a — ib is also a root. COROLLARY. Every equation of odd degree with real coefficients has at least one real root. The roots cannot all be imaginary, else the degree of the equation would be even by the preceding theorem....

...are separately equal to zero. That is to say, Sa+T=0, and S =0. XI, 104] POLYNOMIALS COROLLARY I. An equation of odd degree with real coefficients has at least one real root. COROLLARY II. Any rational integral function with real coefficients may be broken up into real factors,...

...x2 - 12 x + 3. 13. Prove, without assuming the fundamental theorem of algebra, that every algebraic equation of odd degree with real coefficients has at least one real root. 295. Successive Derivatives. The derived function of a polynomial f(x) of degree n is a polynomial...

...x L dy B, C, A positive, BC < AD, Dx=Cy, k = ^ 3. Continuation /dx —==• VO 3 (x) Any polynomial of odd degree with real coefficients has at least one real root . Let x = c be such a root of G 3 (x). Then the integral can be referred to an integral already treated...

...similar to those of problems 26 and 27 for equations of degree 5, and of degree 6. 29. Prove that ever^ equation of odd degree with real coefficients has at least one real root. 30. Show that 2 is a double root of z4 - 4 x3 + 13 x* - 36 x + 36 = 0, and find the other roots. 31....

..._ dy J " TD/1 .4, B, C, A positive, BC < AD, Dx=Cy, k = 4jj C dx 3. Continuation: I Any polynomial of odd degree with real coefficients has at least one real root. Let x = c be such a root of <? 3 (x). Then the integral »i r can be referred to an integral already...

...connected. Exercises 9-15 can all be done using the Intermediate Value Theorem. 9. Show that any polynomial equation of odd degree with real coefficients has at least one real root. 10. Prove that any continuous function /: [a, b]-»[a, b] has a fixed point, that is, a point x such...

...We shall study properties of fidd extensions in Chapter 4. EXERCISES 1. Show that every polynomial of odd degree with real coefficients has at least one real root. 2. Show that any irreducible polynomial over reals is of degree utmost 2. 3. Let /(x) e F[x]. Show...

...(j>(x) and an have the same sign. • 6.11.20 Theorem (Roots of real polynomials). Every polynomial of odd degree with real coefficients has at least one real root. Proof: Let p(x) be a polynomial of odd degree with real coefficients, say p(x) = a0 + a\x + . . . +...

...his proof he reduced the nonalgebraic assumptions to a minimum. He used the following two: I. Every equation of odd degree with real coefficients has at least one real root. II. Every equation of even degree with real coefficients and negative constant term has at least two...