SECTION 4.3 Polynomial Division; The Factor the polynomial function f(x). Then solve the equation f(x) = 0. 39, f(x) =x3 + 4x2 + x-6 40. fx) 5x - 2x 24 41, f(x) =x3-6x2 + 3x+10 42. f(x)-x3 + 2x2-13x + 10 43, f(x) = x3-x2-14x + 24 44.f(x) = x3-3x2 In Ex given. a): Fi b) C in gi - L 二 10x +24ー丁only, this one d) C gi ase 45' f(x) =x4-7x3 + 9x2 + 27x-54 plecs( 46, f(x)...
For the polynomial function f(x) = −5x(x + 2)2(x − 1)3: For the polynomial function f(x) = -5x(x + 2)(x - 1)": 7. (4 points) The leading term when expanded is -5.2". Use this to describe the end be havior of f(x): as r → , f(x) → as I + -00, f(x) → 8. (4 points) Name the zeros of f(x) and each of their multiplicities. 9. (4 points) Come up with a rational function which has y =...
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...
Theorem. Let p(x) = anr" + … + ao be a polynomial with integer coefficients, i, e. each ai E Z. If r/s is a rational root of p (expressed in lowest terms so that r, s are relatively prime), then s divides an and r divides ao Use the rational root test to solve the following: + ao is a monic (i.e. has leading coefficient 1) polynomial with integer coefficients, then every rational root is in fact an integer....
Find the degree 3 Taylor polynomial T3(x) of the function f(x)=(7x+50)4/3 at a=2Find the second-degree Taylor polynomial for f(x)=4x2−7x+6 about x=0thank you! (:
Let F=Z_3, the finite field with 3 elements. Let f(x) be an irreducible polynomial in F[x]. Let K=F[x]/(f(x)). We know that if r=[x] in K, then ris a root of f(x). Prove that f(r^3) is also a root of f(x). Which of the following are relevant ingredients for the proof? If a and b are in Z_3 then (ab)^3=(a^3)(b^3) The Remainder Theorem If a and b are in Z_3 then (a+b)^3=2^3+b^3 For all a in Z_3, a^3=a The first isomorphism...
Let F=Z_3 , the finite field with 3 elements. Let f(x) be an irreducible polynomial in F[x]. Let K=F[x]/(f(x)). We know that if r=[x] in K, then ris a root of f(x). Prove that f(r^3) is also a root of f(x). Which of the following are relevant ingredients for the proof? If a and b are in Z_3 then (a+b)^3=a^3+b^3 If g is an automorphism of K leaves g(r) is a root of f(x) The Remainder Theorem The Factor Theorem...
Part A-Multiple Choice K/U-20 marks] 1. If the leading coefficient of an odd-degree polynomial function is positive, then the function extends from the third quadrant to the first quadrant; that is, as * , y -co and as →-co, y → b. * →-co, y → and as x 0, y →-60 * →-co, y →-co and as x 0, y → d. X-CO, y -co and as x, y →-00 a. c. c. c. 6 2. Which polynomial function...
Using the complex-n-th roots theorem: 5. (a) Use Theorem 10.5.1: Complex n-th Roots Theorem (CNRT) to com- pute all the 4-th roots of -1/4. (b) Factor the polynomial 4x4 + 1 in C[x]. (c) Factor the polynomial 4x4 +1 in R[x]. (d) Use Rational Roots Theorem to prove that the polynomial 4x4 + 1 has no rational roots. Deduce the factorization of 4x4 + 1 in Q[x].
5. Use the Factor Theorem to determine if - 3 is a root of h(x) = x3 – x2 + 27. Do not use synthetic division 6. Write a fourth-degree polynomial f(x) that has roots -2, -5,3 7. Without using your calculator, but by using the change of base equation, show me step by step how to find log, 200 in terms of logarithms with base 10