4/ Let α and β be the roots of ax2 + bx + c--0. Show that the equation whose roots are (α + β)2 and
3(b) Although the polynomial z6-2c4 + x2 + 2 is not a cubic, use theorem 12.3.22 to show that it has no constructible roots. (The idea from this question can be used to do question 2(c)) Theorem 12.3.22: if a cubic equation with rational coefficients has a constructible root, then the equation has a rational root. 3.(c) The following polynomial is cubic but does not have rational coefficiens3. this polynomial (use part (b)) to show that this polynomial has no...
8. Let w cos(2π/5) + isin(2π/5). Here we describe how to express w in terms of square roots. (a) Show that w is a root of the polynomial 24+2+22+21. Hint: 25-1-(-(24+23+22+2+1) (b) Show that w + is a root of the polynomial u2 + u-1 (c) Show that Ve, where V5 means the positive square root of Hint: Figure out the sign of w by adding the polar forms of w and 1/w. (d) Put β--12vS So in part (c),...
Let E = F(a) be a (simple) extension of F. wherea E E is algebraic over F. Suppose the degree of α over F is n Then every β E E can be expressed uniquely in the form β-bo-b10 + +b-1a-1 for some bi F. (a) Show every element can be written as f (a) for some polynomial f(x) E F (b) Let m(x) be the minimal polynomial of α over F. Write m(x) r" +an-11n-1+--+ n_1α α0. Use this...
(4.2) Consider the integral -f 1 J dec 1+3 (a) Show that (4) 1 da 1 +x3 dr 1+r3 (b) Deduce that (3) -re) J f() dr where f is a function to be determined (8) and the (c) Approximate J by means of the three-term Gaussian Quadrature Hint: The roots of the third Legendre polynomial are xo corresponding coefficients for the three-term Gaussian Quadrature are co =,C= , C2= 15 15, 1 0, 32 5 Y 9 (2) (25]...
3. Consider the linear model: Yİ , n where E(Ei)-0. Further α +Ari + Ei for i 1, assume that Σ.r.-0 and Σ r-n. (a) Show that the least square estimates (LSEs) of α and ß are given by à--Ỹ and (b) Show that the LSEs in (a) are unbiased. (c) Assume that E(e-σ2 Yi and E(49)-0 for all i where σ2 > 0. Show that V(β)--and (d) Use (b) and (c) above to show that the LSEs are consistent...
Let α and β be real numbers with 0 < α < βく2m and let h : [α, β] → R>o be a continuous function that is always positive. Define Rh,a to be the region of the (x,y)-plane bounded by the following curves specified in polar coordinates: r-h(0), r-2h(0), θ α, and θ:# β. 3. (a) Show that (b) (c) depends only on β-α, not on the function h. Evaluate the above integral in the case where α = π/4...
write a C programming code for the following prompt. please use stucture concept and test if the code works perfectly. Project Description: In this project, you will write a program to calculate the roots of a quadratic equation. Structure concepts will be used in this project. Your program will prompt the user to enter the coefficients of a quadra coefficientsType. Then it will compute the roots of the quadratic equation and store the result in a structure variable of type...
Solve the Taylor Series. 1. (a) Use the root test to find the interval of convergence of-1)* に0 (b) Demonstrate that the above is the taylor series of f()- by writing a formula for f via taylor's theorem at α-0. That is write f(x)-P(z) + R(x) where P(r) is the nth order taylor polynomial centered at a point a and the remainder term R(x) = ((r - a)n+1 for some c between z and a where here a 0. Show...