+4x Criterion to show that f(x + 1) is irreducible and applying Exercise 12.
17. The real number a = cos 20° is a root of the irreducible polynomial f(x) = 4x? – 3x 3x = in Q[x]. Let E = Q[cos 20°). Show that f(x) splits in E.
Example 4.2.4 shows f=x^n+px+p with p prime implies that f is irreducible over Q by Eisenstein criterion Exercise 1. Lemma 4.4.2 shows that a finite extension is algebraic. Here we will give an example to show that the converse is false. The field of algebraic numbersis by definition algebraic over Q. You will show that :Ql oo as follows. (a) Given n 22 in Z, use Example 4.2.4 from Section 4.2 to show that @ has a subficld such that...
-----..on 12. The functions (*) s f(x) = 3x - 1 fy(x) = 4x and f(x) = 2x are linearly dependent. Show this by finding values of CC, and c,, not all zeros, such that ºf(x) + c, f(x) + e, f,(x) = 0.
Prove that x4 + 3x + 4x² + 8x + 11 is irreducible in Q[x] . Make sure to completely justify all your claims.
Prove that the polynomial 9x^4 + 4x^2 − 3x + 7 is irreducible in Q[x].
Exercise 5. Extreme values (8 pts+12 pts) Let f(x,y) = 2x2 - 4x + y2 – 4y +1. 2) The point (1,2) is: a. a local maximum for f b. a local minimum for f c. a saddle point for f O a. b. O c.
Show that the following polynomials are irreducible over Q. (a) (8 points) f(1) = 5.rº – 1826 + 30x4 – 6r2 + 12x + 60 (b) (12 points) g(x) = r" - 6.12 – 4.: +3
Theorem 14.7. If f(x) € R[x] is an irreducible polynomial, then deg(f(x)) is either 1 or 2. We can determine which quadratic polynomials in R[x] are irreducible by using the quadratic formula and checking for real roots. Activity 14.8. Factor f(x) = 2 – 4.x in R[2] into a product of irreducible polynomials in R[2].
Exercise 5. Extreme values (8 pts+12 pts) Let f(x, y) = 2x2 - 4x + y2 – 4y +1. 1) The number of critical points of f is: a. 0 b. 1 c. 2 d. 3 2) The point (1,2) is: a. a local maximum for f b. a local minimum forf c. a saddle point for f
please show answer in full with explanation, also show matlab 1. Consider the function f(x)2.753 +18r2 21 12 a) Plot the graph of f(x) in MATLAB and find all the roots of the function f(x) graphically. Provide the code and the plot you obtained. b) Compute by hand the first root of the function with the bisection method, on the interval -1; 0) for a stopping criterion of 1% c) How many iterations on the interval -1, 0 are needed...