In real analysis,Prove that if A is countable, then m(A) =0.

Q: Find necessary and sufficient conditions on the numbers a, b, c E R under which the polynomial xª +…

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Q: . Do the polynomials x? + 2, 2x? – x + 1,x + 2, and x² + x + 4 span P2?

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Q: 4. In each part, determine whether the polynomial is a linear combination of P, = 2+x+x², P, = 1-x²,…

A: We have to determine whether the polynomial is a linear combination of given polynomials.

Q: Show that Q(i) is a subfield of c

A: Subfield of a field

Q: On what intervals is the polynomial p(x)= 2x –15x² +36x+50.concave down? Concave up?

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Q: 1. Show whether the given polynomials 1+4x + x², 2+ 5x + 3x2, 3x 5x, and 3+ 15x - 8x spans p,.…

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Q: Let P3 have the inner product given by evaluation at -7, -1, 1, and 7.

A: Given, ℙ3 have the inner product given by evaluation at -7, -1, 1 and 7. And P0(t)=3, p1(t)=2t,…

Q: Evaluate the quadratic form f(x) = xT Ax for the given A and x.

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Q: Find the coefficient of x²y² in the expansion of (–x+ 3y)*.

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Q: Find a polynomial P(x)=x3+ax2+bx+c satisfying ALL of the following properties:

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Q: Find all values of k such that f (x) is divisible by the given linear polynomial .

A: Given polynomial is fx=k2x3-4kx+3. To find all values of k such fx is divisible by x-1. Solution:…

Q: Do the polynomials x? + 2,2x2 – x + 1,x + 2, and x² + x + 4 span P2?

A: Let u1 = x2+2, u2 = 2x2-x+1, u3 = x+2, and u4 = x2+x+4. Notice that u1 + u3 = u4. So, u1, u2, u3,…

Q: Test the polynomial x – 2x3 + 6x – 3 for irreducibility over Q.

A: We have to check whether the polynomial x4-2x3+6x-3 is irreducible over the set of rational numbers…

Q: Explain the formula for the Hermite interpolating polynomial?

A: In any polynomial function if the values for the function is known for some data points and the…

Q: Find the coefficient of x5 in (1 – 2x)-7 using generating function.

A: Given to find the coefficient of x5 in 1-2x-7 The generating function will be Gx=1-2x-7…

Q: (b) Expand and simplify in ascending powers of x as far as the term in x" 1+3x

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Q: If b is a zero of x2 + x + 2 over Z5, find the other zero.

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Q: How do i find the ral seros of  a given polynomial and their corresponding multiplicities?…

A: We have to find the zeroes and their multiplicities.

Q: Given that f(x)=2x+1, and g(x)=x²-x+3. Determine (gof)(x), expand, and simplify.

A: From the functions..

Q: Show that if p is a positive integer, then the binomialseries for f(x) = (1 + x) p is a polynomial.

A: The binomial expansion is given as 1+xn=1+nC1x+nC2x2+nC3x3+.....nCnxn

Q: Write the following polynomial as a preduct of ivreducibles in Q [x], R[x] and C[x] x+*+ 5x-22

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Q: Q3: Factorize f (x) = 2x* + x³ + 2x + 4 € Z5[x] as a polynomials product to irreducible over Z5?

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Q: Express the following polynomials by using sigma notation.   x + x2 + x3 + x4

A: Given polynomial expression is: x + x2 + x3 + x4

Q: For f(x)=2x4 +2x3-1 and g(x)=2x3 +2x+3 in Z4[x], the degree of fg is 7. True or false

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Q: Evaluate the quadratic form f(x) = xT Ax for the given A and x.

A: We need to  Evaluate the quadratic form f(x) = xT Ax

Q: AE Mmxn amed suppose be non Let that A=xY is the fall ragak factorization that At =x*y then Show

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Q: (a) What is the degree of Q(7, V3) over Q ? Justify your answer. Is the polynomial X - 5 € QIX)…

A: Degree of extension- Let E be an extension field of a field F i.e. F is a subfield of E, also E is a…

Q: Find m so that x– 5 is a factor of f (x)= 4x² – 21x² +mx–15.

A: Consider the given polynomial fx=4x3-21x2+mx-15

Q: Use long division to find a value of k such that f(x) is divisible by d(x). f(x) = x4 + x3 + 3x² +…

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Q: Given the polynomial f(x)=−3x3+10x2+1, what is the smallest positive integer a such that the…

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Q: the integret fector of the bifferentiel e qration →き x28x he is 2. O Not found

A: Consider the given differential equation xdy-ydx=x2y3dx and rewrite it again by doing…

Q: Given that f(x)=x^2-3x-54f(x)=x2−3x−54 and g(x)=x+6g(x)=x+6, find (f\cdot g)(x)(f⋅g)(x) and express…

A: fx=x2-3x-54gx=x+6

Q: A farmer’s rectangular field has an area that can be expressed as thetrinomial x2 + 2x + 1. In terms…

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Q: Give the smallest extension of Q that contains all the zeros of (x4 – 8x² + 15)(x² + 1). -

A: Yes

Q: What is the fermat value, in polynomial form, of integer 'a', if a=x^4 + 1; p=x^2 + 1?

A: We have to find the Fermat value 'a' which is given in the following form, a=x4+1p=x2+1

Q: Lonstant Find the Valre or k fed ahich thhe $yotem of X, + Xx +Xg =/ 3x-X,-Xg = 4 2,+ Sx+5x3ニド…

A: The objective of the question is to find the value of k for which the system of equations is…

Q: Show that Z is not isomorphic to Q.

A: We can prove this by the method of contradiction. Assume that there exists an isomorphism ϕ:ℚ→ℤ.…

Q: Find the conditions that the zeroes of the polynomials Ax) =x – px+ qx – r may be in arthametic…

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Q: The interpolating polynomial p(x) of degree 3 of f(x) = 6x4 – 2a3 + 9 is p(x) = -23 + 9. Select one:…

A: From interpolation formula we know that every polynomial f(x)of order n is approximated by a…

Q: Consider the polynomial f(x) = 25x - 9x* + 3x? - 12 %3D show that it is irreducible in Q[x]

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Q: Use synthetic division to show that c is a: a. P(x)=x'-4x² + 5x – 2; c = 2

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Q: According to the fundamental theorem of algebra, how many zeros does the function f(x) = 4x³ – x2 –…

A: Fundamental theorem of algebra The fundamental theorem of algebra states that- Polynomial P(x) or…

Q: Using the nodes (xo=2, x;=2.5 and x; =4), to find the second interpolation polynomial for f(x) =

A: According to the given information, it is required to find the second interpolation polynomial for…

Q: i Yi -2 3. -1 x2-9x+8 3. 1. 1, 2.

A: Solution is uploaded here

Q: 6x-C), Prove that If2zf(x+ct)+Ø て

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Q: where x3 = 5x1 + 8x2 [x3] W : X: x = |X2 X-

A: The given subset W is: W= x :  x=x1x2x3    where  x3=5x1+8x2

Q: Show that Z[√−D] is not a unique factorization domain, where D is squarefree and positive

A: Given: To be prove Z-D is not unique factorization domain.

Q: determine if the statement is true or false. If a statement is false, explain whyThe sum of two…

A: Given:The sum of two polynomials each of degree 5 will be less thanor equal to degree 5

Q: Consider a quadratic polynomial with a negativ main coefficient and a negative discriminant. Which…

A: Consider a quadratic polynomial with a negative main coefficient and a negative discriminant. Which…