Show that f(x) = x³ + 2x + 20 is irreducible over Q.

Q: Find the antiderivative F of f(x)=4-3(1+x^2)^-1 that satisfies F(1)=-2.

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Q: Is f(x)=2x+4 reducible over Z ニ

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Q: i. Suppose p is an irreducible polynomial of degree n with coefficients in Q. Then p has

A: Determine whether p(x) = x3+x+1 is irreducible over  Sol: we know that , let f(x) be a polynomial in…

Q: Show that f(r) = O(|x – xo|*) and g(x) = o(]x – roP) imply f. g(x) = o(|x – xo]k+i). - -

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Q: Show that : f(x) = x2 + x + 2 e Zg[x} is irreducible. Further, find the order of f (x).

A: Definition: 1. A polynomial f(x) ∈ ℤp[x] is said to be irreducible if and only if it has no root in…

Q: For which value of x does the Maclaurin polynomial Tn satisfy Tn(x) = f (x), no matter what f is?

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Q: Let f(x,y) = x - 2xy – y“. Compute f(3,0) and f(3, – 7).

A: To compute f(3,-7) Given:- f(x,y)=x2-2xy-y2

Q: Form the interpolating polynomial of degree k that satisfies pG) (xo) = f(G) (xo) for 0<j< k.

A: To form: The interpolating polynomial of degree k that satisfies pjx0=fjx0 for 0≤j≤k

Q: Let g=yf(xy). Show that: - дд у ду дд - Х = g дх -

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Q: Let f(z) - 4z-1 pnd alz) - ar+b, where a pnd b are positive constants. If the composite

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Q: Prove that f(x) = [[sin²(2log r)– 3 sin ( 2logt)+2] dt %D 1. Cannot be decreasing any where,

A: To prove that f(x) can not be decreasing anywhere, where f(x) is defined as…

Q: Show that the nth Maclaurin polynomial for f(x) = e* is Tn (x) = 1+ 2! n!

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Q: Use the intermediate value theorem to show that f(x) = x^4 + 8x^3 −x^2 + 2 has a zero in I = [−1,…

A: Given,

Q: ) Find the Taylor polynomial p2 for f(x) = at x = 2. x3

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Q: Compute the Maclaurin polynomial of degree 4 for the function f(x)=cos(x) ln(1-x) for -1<x<1.

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Q: 1. Find of ахду OF f(x,y) = xy (x+y-1) -x2

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Q: Let f(æ) = – 3x* In æ %3D f'(x) = f'(e²) =

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Q: (d) Is f invertible? If so, what is f-?

A: f:ℝ→ℝ is defined as  fx=x1+x⇒fx=x1-x, x&lt;0x1+x, x≥0 f is both 1-1 and onto. Hence, f is…

Q: Apply Newton's forward formula to obtain a third-degree polynomial that interpolates y = 2 cos(nx)…

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Q: Consider the following. f(x) = x 4 1 4 1 3 - x2 %D - Find the critical numbers. (Enter your answers…

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Q: Find the linearization L(x) at x =a. .3 f(x)=4x² +3x-1 a =-3 L(x) =

A: The linearization of a differentiable function f at a point x=a is the linear function…

Q: (d) If f(x) is irreducible over Q then f(x+ a) : a e Z is irreducible over Q.

A: Use the property of reducible to start the proof.  According to the property of contrapositive, "if…

Q: In Exercises 1-8, find the general antiderivative of f and check youranswer by differentiating. f…

A: Given     fx =9x+15x-2

Q: sin

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Q: Compute the least squares quadratic polynomial p2(x) which gives the best fit to e x over the…

A: We need to find two things, A least square quadratic polynomial p2(x) which gives the best fit to…

Q: Show that the Maclaurin polynomials for f (x) = ln(1 + x) are

A: The given function is fx=ln 1+x. fx=ln 1+x f0=0 f'x=11+x f'0=1 f''x=-11+x2 f''0=-1…

Q: Find the linearization L(x) at x= a. f(x) = 2x° + 5x - 3 a= 3 L(x) =

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Q: Consider the cubic functionf (x) = ax3 + bx2 + cx + d, where a  0. Show that f canhave zero, one,…

A: We have fx=ax3+bx2+cx+d This implies  f'x=3ax2+2bx+c Set f'x=0 i.e 3ax2+2bx+c=0 and solve it using…

Q: Determine whether the function 2х + 1 f(x) %3D 2r2 2x2 – x – 1 is continuous: a. at 2; b. at 1.

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Q: Prove, if f(x)= V1+x² –In +1+ 1 is increasing and concave on x X E (0;+∞) .

A: f is increasing if first derivative is positive in the interval  and f has concave if second…

Q: what is the antiderivative F(x) of f(X) = 3x1/2-x-1 that satisfies the condition  F(4)= 0? A,…

A: The given function is fx=3x12-x-1 that satisfies the condition F4=0. Find the antiderivative of Fx.

Q: . Suppose that f is a polynomial of degree 1 and s1, is the linear spline that interpolates f at the…

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Q: The Taylor polynomial of degree 3 centered at x=0 of the function f defined by f(x)=sen(8x)+cos(9x?)…

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Q: b) Show that the function f(x) = x7 +1 is invertible %3D

A: Given that  f(x)=x^7+1

Q: Find the Taylor polynomial p2 for f(x) = cos(2x) at x = 5.

A: Given that f(x) = cos(2x) at x= π/8  Find the taylor polynomial p2 . Here i attached solution of…

Q: Suppose that f (0) = 2 and f' (x) ≤ 3 for x &gt; 0. Apply the MVT to the interval [0, 4] to prove…

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Q: (c) Show that : f(x) = x2 + x + 2 e Zg[x] %3D is irreducible. Further, find the order of f (2).

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Q: Find the linearization L(x) at x = a. f(x) = - 4x3 - 3x+1 a = - 3 L(x) =

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Q: Show that the function f(x) = x’ + 1 is invertible from R to R.

A: First prove that the  function is one -one  let f(x)=fyx7+1=y7+1x7=y7x=y therefore, the function is…

Q: 10. If f(x) = x³ – 3x +1 is defined on [0,1], then by using False position method, find the first…

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Q: - Let Q(x) be a polynomial of degree less than N. Prove that Q(x)PN(x)dx = 0.

A: Note, that this result is a specific case of orthogonality of Legendre’ s polynomials. We will prove…

Q: Q3: a) Let f : Z → Z be such that f(x) = Vx² – x + 1. If invertible, and if it is, what is its -…

A: the function is invertible if its one -one and onto. The function is one - one if f(x1)=f(x2)…

Q: Use Part 1 of the Fundamental Theorem of Calculus to find the g'(x) if: g(x) = 1 +t² dt 3x2

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Q: | Are fi(x) = x and f2(x) = x² orthogonal on [-2, 2] inter- val? Find c and c2 so that f3(x) =…

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Q: Let f(x) = x² In x, where x > 0. Show that f(x) has only one critical point.

A: Given data: The given function is f(x)=x2lnx. The critical point is the point where the slope of the…

Q: In Exercises 1-8, find the general antiderivative of f and check youranswer by differentiating. f…

A: Find the general antiderivative of f and check youranswer by differentiating. f (x) = 18x2