Determine whether the following polynomials are irreducible in Q[x].
(i) f(x) = 3x2 – 7x – 5
(ii) f(x) = 2x3 – x – 6
(iii)f(x) = x3 + 6x2 + 5x + 25
Determine whether the following polynomials are irreducible in Q[x]. (i) f(x) = 3x2 – 7x –...
Write the polynomial f(x) as a product of irreducible polynomials in the given ring. Explain in each case how you know the factors are irreducible. 1) f(x) -x* + 2x2 +2x 2 in Z3[x]. 2) f(x)4 + 2x3 + 2x2 +x + 1 in Z3[x]. 3) f(x) 2x3-x2 + 3x + 2 in Q[x] 4) f(x) = 5x4-21x2 + 6x-12 in Q[x)
1. (15 points) Prove whether the following sets are linearly dependent or independent, and determine whether they form a basis of the vector space to which they belong. s 10110 -1 ) / -1 2) / 2 1 17 ) } in M2x2(R). "11-21 )'(1 1)'( 10 )'(2 –2 )S (b) {23 – X, 2x2 +4, -2x3 + 3x2 + 2x +6} in P3(R) (the set of polynomials of degree less than 3. (c) {æ4—23+5x2–8x+6, – x4+x2–5x2 +5x-3, x4+3x2 –...
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
Problem 4. Consider f(x) = x5+ x4 + 2x3 + 3x2 + 4x + 5 ∈ Q[x] and our goal is to determine if f is irreducible over Q. We compute f(1), f(−1), f(5), f(−5) directly and see that none of them is zero. By the Rational Roots Theorem, f has no root in Q. So if f is reducible over Q, it cannot be factored into the product of a linear polynomial and a quartic polynomial (i.e. polynomial of...
10) Determine whether the matrix operator is invertible, if so, find its inverse. a)T(x, y) = (3x + 4y, 5x + 7y) b)T(x1, X2 X3) = (x; + 2x2 + 3x3, xz – X3, X; +3x2 + 2x3)
Differentiate the following function. f(x) =e 32 - 7x (e-3x2 +7x) = dx
Describe the end behavior of given polynomials. Sketch the behavior of the branches 5. f(x) =-3x3 +7x2 - 2 an 6. f(x) = -5x6-2x3 + x 7. f(x) = 4x4 - 3x2 + 2 8. f(x) = 2x + 3x2 - X
[6 points] Suppose that f'(x) = 3x2 + 2x + 7 and f(1) = 11. Find the function f(x). Of(x) = x3 + x2 + 7x + 11 Of(x) = x3 + x2 + 7x + 2 Of(x) = 6x + 5 Of(x) = x3 + x2 + 9 10. V MY NOTES [6 points] Find the integral: 15x3/2 + 21n|x[ + c 10 x3/2 + 2\n\xl + C O 5x1/2 - + c şx-1/2 - + c
x+5 + 4. Solve +7x+2 x-1 212+5x+2 3x28x+4 (a) You know the drill! Factor the denominators! (NOTE: If you need help factoring these polynomials, see Helping Handout: Lab 1B) i. Factor the first denominator: 6x2+7x +2 = ( OC ) ii. Factor the second denominator: 3x+8x+4-( iii. Factor the third denominator: 2x2+5x+2 = ( ) (b) Rewrite with factored denominators: x+5 x-1 X + (2x+ 1)(3x+2) x+2)(3x+2 ) (c) Find the restrictions: (x+2) (2x+1) AND AND (d) Find the LCM:...
Section 1.3 3. For each of the following, determine algebraically whether function is even, odd, or neither: A) f(x) = 5x - 7 Answer B)f(x) = 8x - 7x + 10 Answer c) f(x) - 6x2 + 1 Answer