Investigation 10.31. Let f(x) be a polynomial. Consider the construction (a) Is always a polynomi...
Consider the polynomial f(x) = x+ + 4x + 4x? a. What is the degree of this polynomial?_ b. What is the y intercept?_ c. What are the roots (zeros) of this polynomial? d. What is the end behavior of this polynomial ? e. Sketch this polynomial?
9. Let f(x) = sin(x). (12 marks) In the following we will consider its Taylor Polynomial and its Taylor Series. You can assume that the Taylor Series converges, no need to prove it. (a) (4 marks) What is the Taylor polynomial of degree 9 centred at 0 for f(x)? Justify your answer pg(x) = (b) (4 marks) Approximate the integral (sin(x3) dx Jo using your answer from (a). Justify your answer.
6) a) For the polynomial f(x) = 4.73 - 7x +3, check that 1 is a root. b) Use the Factor Theorem to find all other roots and their multiplicities. 7) Use the Rational Root Theorem to find all roots of f(x) = 4.3 - 3x + 1.
4. Describe how to find the possible rational zeros of a polynomial function. Use the function f(x)-2x +13-9x2 a. List all possible rational roots b. Use Descartes's Rule of Signs to determine the possible number of positive and negative f(x) = 2t to answer the following. real roots c. Use synthetic division to test the potential rational zeros and find an actual zero d. Use the quotient from part(c) to find all the remaining zeros e. Rewrite f(x) in completely...
5. Let F be a field, and let p(x) ∈ F [x] be a separable, irreducible polynomial of degree 3. Let K be the splitting field of p(x), and denote the roots of p(x) in K by α1, α2, α3. a) (10’) If char(F ) does not equal 2, 3, prove that K = F (α1 − α2).
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....
2. (10) Let p be an odd prime. Let f(x) E Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D2p of a regular p-gon. Prove that f(x) has either all real roots precisely one real root or 2. (10) Let p be an odd prime. Let f(x) E Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D2p of a regular p-gon. Prove that f(x) has either...
Consider the function: f (x) = b - sin(x), where bis an arbitrary number. Are roots always possible for any value of b? Yes No Consider the function: f (x) = a - tan(x), where a is an arbitrary number. Are roots always possible? True False
Factor the polynomial f(x). Then solve the equation f) o. 10) f(x) x3+5x2- 9x-45 State the domain of the rational function. (6 points) 11) g(x) =ー2 x +2 Given that the polynomial function has the given zero, find the other zeros. 12) f(x)=x3-4x2 + 9x-10:2