Use Liouville’s theorem (Problem 3 ) to prove the fundamental theorem of algebra (see Problem 7.44). Hint: Let
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- 4. Consider the polynomial f(x) = x* + 1. (a) product of two irreducible quadratics. Explain why ƒ has no real roots, and why this means f must factor as a (b) Factor f and find all of its complex roots.arrow_forwardThe problem is from Linear Algebra Done Right by Axler. Could you teach me how to do this in detail?arrow_forwardIf q(x) is an arbitrary polynomial in Pn it follows that q(x) = c0p0(x) + ... + cnpn(x) (1) for some scalars c0, ... , cn (a) Show that ci = q ( ai) for i = 0, ... , n, and deduce that q(x) = q (a0)p0(x) + · · · + q( an)pn(x) is the unique representation of q(x) with respect to the basis ẞarrow_forward
- Solve the following problems and show your complete solutions. Write it on a paper and do not type your answer.arrow_forwardConsider the following quadratic form f : R3 → R, function is in the image Find the values a ∈ R for which this quadratic form is:1. positive definite2. positive semidefinite3. negative definite4. negative semidefinite5. indefinitearrow_forwardAs mentioned in class on Tucsday, the Legendre polynomials are a set of polynomial functions defined as follows: Po(r) = 1 P(r) = 1 1 P2(r) – (3r – 1) P{(#) = () 2! da: d (교2 We claimed without proof that the inner product (P(r)|Pm(x)) = L', P(x)Pm(1)dr = 2/(21 + 1)dim, which means that the Legendre Polynomials are orthogonal but not orthonomal. (a) Show through explicit computation that the inner product (P (r)|P2(x)) = 0 and that the norm of P(1) is V2/(2 + 1+1). (b) Show through explicit computation that the inner product (P(r)|P;(1)) = 0 and that the norm of P3(r) is V2/(2 + 5+1). Hint: Use Mathematica or Wolfram Alpha to find P3(x) and P;(r).arrow_forward
- 1. Let V = P₂(x) and define = 2ƒ(0)g(0) + f(1)g(1) +3ƒ(2)g(2). (a) Prove that this is an inner product (You may use the fact that a quadratic poly- nomial with three roots must be zero). (b) Let f(x) = 1 + x and g(x) orthogonal to f. = x² - 3. Compute Projƒg and the projection of garrow_forward3. Recall that a quadratic polynomial has at most two roots. Show, using Rolle's theorem, that a cubic polynomial (i.e. f(x) = ax³ + bx² + cx+d with a, b, c, d real numbers, and a + 0) can have at most three real roots. Hint: what does Rolle's Theorem tell you about the relationship between the zeros of f(x) and f'(x)?arrow_forwardQuestion 6. Prove by using elementary argument that any polynomial with real coefficients of degree can be represented as a product of linear terms (z − a), where a is a real (constant) number and quadratic terms (z² − bz + c), where b, c are also real constants. [Hint: you may assume that roots of polynomials with real coefficients are either real or come in conjugate pairs, i.e. if p(x + y) = 0, then p(x − iy) = 0.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningAlgebra for College StudentsAlgebraISBN:9781285195780Author:Jerome E. Kaufmann, Karen L. SchwittersPublisher:Cengage Learning