Let u be a root of ƒ = t³ − t² +t+ 2 € Q[t] and K : = (a) Show that f = mo(u). (b) Express (u²+u+1)(u² Q(u). −u) and (u−1)−¹ in the form au²+bu+c, for some a, b, c € Q.

Q: Let R={(x,y) ERxR: |x| = |v|). Then R a function on R. оо True False

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Q: Show that the operator U is positive U: 1² ([0,1]) → L²([0,1]), Uƒ (x) = X[{0,1] [~*\f(t)\dt

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A: To solve the following

Q: Let V = span{e2", xe2", x²e?=}. (a) Show that (a1e2a + a2xe2" + a3x²e2") e V for any a1, a2, az E R,…

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Q: 4. A) Factor f(x) into linear factors given that k is a zero of f(x). B) Find all roots of the…

A: f(x)=6x3+19x2+8x-5, k=-1To:a) Factor f(x) into linear factors b)Find all roots of the function

Q: Show that the set of functions 1,cosx,sinx,cos2x,sin2x,…,cosmx,sinmx,… is orthogonal on the interval…

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Q: Suppose f: X→Y, and A, B CY. Prove that (a) f-¹(AUB) = ƒ−¹(A) U ƒ−¹(B). (b) f-¹(ANB) = f¯¹(A) n…

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Q: Let f: Z → Z be defined by f(x) = x² + 1, and let C = {1,2,3}.

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Q: Using the substitution u = (A) (B) C D 1+1 du [₁ -1/2 du [(u ¹/2 - u-1/2) d du (u - 1) / u-²1) x + 1…

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Q: u = [2, 6, –5] and v= [-3,–1,6] %3D A · A + (A + ng-) · (A x n)

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Q: 6. Prove that |xy| = |x|ly| for all real numbers x, y.

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Q: (2) Let f(x) = ax³ + bx? + cr + d, where a, b, c, d e R and a + 0. Prove that f(x) must have a root.…

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Q: 2. Let us define a. Verify that f(x, y) = c. Now consider the function git Similarly, consider the…

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Q: Show that f : ℝ → ℝ, f(x) = 5x − 3 is a bijective function.

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Q: c) Which of the following functions are analytic in 121² 2 2: 42²26649 (ii) Z²-1 Z-√√Z²74

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