I am trying to find out if the polynomial 12x^4+8x^3+9x^2+4x+4 is reducible or irreducible over Q. We can't apply Eisenstein's criterion here. But we can try this: 12x^4+8x^3+9x^2+4x+4 = x^4+(2/3)x^3+(3/4)x^2+(1/3)x+(1/3) x^4+(2/3)x^3+(3/4)x^2+(1/3)x+(1/3)=(x^2+ax+b)(x^2+cx+d) = x^4+(a+c)x^3+(b+d+ac)x^2+(ad+dc)x+bd We get the equation system: a+c=(2/3) b+d+ac=(3/4) ab+bc=(1/3) bd=(1/3) I get the following result: b=(1/2) d=(2/3) c=(2+sqrt(19))/6 and (2-sqrt(19))/6 a=(2/3)-((2+sqrt(19))/6) and (2/3)-((2-sqrt(19))/6)  The expression in a contains an irrational number, sqrt(19), this means that the polynomial is irreversible over Q. Is this correct?

Q: The fourth-degree polynomial f (x) = 230x* + 18x³ + 9x² – 221x - 9 has two real zeros, one in [-1,…

A: Bisection method: This method is also called the interval halving method and it is used to find the…

Q: 7.) Explain why the eror term Rn(x) is ficitely bonded in every Tayler polynomial. Asnincreases, why…

A: Nothing

Q: polynömial that passes through the three points (0, 1), (1, 2) and (2, 23). If the coefficient of x…

A:

Q: Suppose ƒ(0) = 1, ƒ'(0) = 2, and ƒ"(0) = -1. Find the quadraticapproximating polynomial for ƒ…

A:

Q: Roots of 1, 2, -4 y-in

A: Polynomial: We need to find a polynomial whose degree is even. Roots are 1,2,-4 y=intercept is -36…

Q: Why are special methods, such as l’Hôpital’s Rule, needed to evaluate indeterminate forms (as…

A: l’Hôpital’s Rule are needed to evaluate indeterminate forms

Q: The polynomial function f(x) = x² – 3x – 5 has two real zeros. |- (A) Determine (without iterating)…

A:

Q: Use a Taylor polynomial with 4 nonzero terms to estimate the value of the integral. 1 evxdx

A:

Q: Find an approximation to V11 using 4 steps Bisection Algorithm

A:

Q: Write the finite difference approximation of utt- Uxx 0 used to solve parabolic PDES. %3D

A: To write the finite difference approximation of - utt-uxx=0—(1) utt is given as-utt=ui,j+1-uijKand…

Q: SV21+4:-: dz

A:

Q: Find ALL irreducible polynomials of degree 2 or 3 in Zr].

A: An irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the…

Q: 6- Show that the Polynomial has a Z4 [x] 2X+1 in multiplicative invers. in 24 [x].

A: Given

Q: Form a polynomial with a Degree 4; zeros:4+2i, -1 multiplicity 2

A:

Q: 1. Use polynomial fitting to find a closed form for 5, 8, 13, 20, 29, ...

A: Given , Sequence = 5,8,13,20,29,.....   To find: Closed form of sequence i.e. formula for nth term .

Q: Form the polynomial equation with integral coefficients whose only roots are given. 3,4,2

A: if a is the root of the polynomial then (x-a) will be the factor of the polynomial  And we can write…

Q: The fourth-degree polynomial f (x) = 230x* + 18x³ + 9x² – 221x – 9 has two real zeros, one in [-1,…

A: Iteration formula for secant method is:  xn+2=xn-f(xn)⋅xn+1-xnf(xn+1)-f(xn)…

Q: Select the factors that multiply to create a polynomial with this description: (1) As x→-0, y→-∞ (2)…

A: Given x→-∞ and y→-∞ and x→∞ and y→∞ and x=4 is a root

Q: a) Use the bisection method to approximate a zero of f(x) = x³ – 3x² +1 with an error of less than =…

A: We have to use bisection method to approximate a zero of the function  f(x) = x3-3x2+1

Q: Write a polynomial f (x) that satisfies the given conditions. 1 1 (multiplicity 1) and with f(0) =…

A: f(x)=ax+15x+15x-12 f(0)=a·15·15·-12 1 = -a50 a = -50

Q: (b) Show that factorial polynomial corresponding to P,- k³ – 4k* + 2k³ – k² + k + 10 is P, = k(5) +…

A: Given: The given polynomials is Pk=k5-4k4+2k3-k2+k+10, To prove Pk=k5+11k4-k3-9k2-k1+10

Q: Express p (x) = 1 + 2x -5x2 as a Taylor polynomial about a= 1

A:

Q: Determine the first root of: f (x) = –13 – 20x + 19x2 – 3x³ with bisection method using initial…

A:

Q: For the polynomial Ax) = 13(x – 6)(66 – 37x²) (a) Report the exact values of the zeros. 66 66 -6 37'…

A:

Q: . Find the Taylor polynomial p3 centered at a = 4 for ƒ(x) = 2/x. %3D

A: To find the Taylor polynomial p3 centered at a=4 for the given function f(x)

Q: 6. Verify whether two polynomials with real coefficients Xª+X-2 and X3+2X –3 are relatively prime or…

A:

Q: 2. Identify the approximate input values of any relative maximums and minimums of this 5th degree…

A: I am going to solve the given problem by using some simple algebra to get the required result.

Q: The fourth-degree polynomial f (x) = 230x4 + 18x³ + 9x² – 221x 9. - - has two real zeros, one in…

A: The given polynomial is - f(x)=230x4+18x3+9x2-221x-9 It is also given that the given function has…

Q: (This is closely related to "Taylor polynomials" which appear at the end of 1st year calculus.)

A:

Q: Find a polynomial Junction of lenst degree having only reil Coejficients with 3eros as giver 2

A: We need a polynomial having real coefficients with zeros at 2 and 1+i. We know that any non real…

Q: Find a Polynomial with integer Coeffcients that Conditions. given Satisfies 'Ithe P has degree 3 and…

A: Given, Polynomial has degree 3 and zeros 5 and i

Q: 11. Summarize the Intermediate Value Theorem for Polynomials in your own words.

A:

Q: 1.) Determine the absolute and relative error when approximating 2x8 using sixth-degree Taylor…

A:

Q: Find the Taylor polynomial of degree 4 about a = 1 for f(x) = 3/x. Write out the 1st 4 nonzero…

A:

Q: Must the values be ordered from smallest to largest for the method to findNewton polynomials to…

A:

Q: Find the best (in the sense of least squares over (-1,1)) second degree polynomial approx- imation…

A:

Q: Express x8 - x as a product of irreducibles over Z2. Please be clear, correct and legible. Thank…

A:

Q: A Cubic polynomial is dlefined by the table 3. 10 tind the equation of the palyinmal Using Lagrange…

A:

Q: Use Bisection method to solve (x - 2)¯ - In (x) = 0, [1, 2]

A: Please refer to the image below

Q: Derive interpolating polynomial for log₁0 1=0, log₁, 2=0.3010, log, 3=0.4771, and hence calculate…

A: Disclaimer: Since you have asked multiple questions, we will solve the first question for you. If…

Q: +∞ x sin(x2) : = - Σ n=0 (-1)" .4n+3 (2n + 1)! X

A: Let's find 7th degree maclaurin polynomial.

Q: Suppose you use a second-order Taylor polynomial centered at 0 to approximate a function ƒ. What…

A: Polynomial: Given that we are approximating the function f by using Taylor Polynomial with n=2. Now…

Q: 2)Let PCv) =x" -5x 74 A. find all Possible rotional Zeros b, find Compiete factorizatin Of p C. find…

A:

Q: # (u²-1)4 (u² + 1)4 du

A:

Q: Suppose a polynomial of degree 4 with rational coefficients has the given numbers as zeros. Find the…

A: Zeros of a polynomial: Let the other zero be kThen the equation of the polynomial…

Q: Give one example of a polynomial of degree that has a zero at x=5 with multiplicity 2

A: zero at x=5 with multiplicity 2

Q: Use syntheti¿ division to firid fiz) in f(x) = -x+5x²-7x+1

A: We have to find function value

Q: Suppose that a polynomial function of degree 4 with rational coèfficients has -2 +51, 5-5 as zeros.…

A: Complex root :  The complex conjugate of a complex zero must also be a zero . So if you have any…

Q: 24. Determine the number of zeroes of the polynomial z7 – 6z5 + 8 in the annulus 1 < |z| < 2.

A:

Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.