### Solve: Simplifying Polynomials

Consider the following expression: Polynomial: (-5t^2 + 2t + 5) + (-8t^2 + 6t) - (9t^2 - 2t -3) Make sure to show all of your work.

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Consider the following expression: Polynomial: (-5t^2 + 2t + 5) + (-8t^2 + 6t) - (9t^2 - 2t -3) Make sure to show all of your work.

Please answer the following: Express 81 as a product of prime numbers.

Please help with the following: Express 36 as a product of prime numbers.

Showing all of the steps which are involved, express 88 as a product of prime numbers.

Prime number factorization Express 45 as a product of prime numbers

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Let p and q be prime number greater than 3. Prove that 24|p^2-q^2

Problem 1: Prove that there are no integers x, y, and z such that x^2 +y^2 + z^2 = 999 Problem 2: Show that square root of 2 cubed is an irrational number. Problem 3: For each of the following pairs a and b, use the division algorithm to find quotient q and remainder r. (a) b=189, a=17

#1: f(x)=(1x^3-18x^2+101x-168)/(1x^3-2x^2-41x+42) Find: All Roots:? All Holes:? All Vertical Asymptotes:? The answer to the 1st question is NOT 8 or -8. #2: f(x)= (-2x^3+37x^2-222x+432)/(1x^3+3x^2-64x-192) I need the roots: (I have two of them (9/2) and (6) but I need the third)? I need to know the hol

Section 4.3 Collect like terms. 4 4 52. 3a - 2a + 2a + a Collect like terms and then arrange in descending order. 3 3 4 66. -1 + 5x - 3 - 7x + x + 5 Classify the polynomial as a monomial, binomial, trinomial, or none o

1- For n belongs to N (set of natural numbers) let B(n) denote the number of digits used in the binary representation of n. For example B(1) = 1; B(2) = 2; B(3) = 2; B(4) = 3: Find a closed formula for B(n) for an arbitrary n belongs to N. 2: Prove that if gcd(a, b) = d then a/d and b/d are relatively prime. 3- Find

Prove that F1F2 + F2F3 + F3F4 + ...+ F2n F2n+1 = (F2n+1)^2 - 1.

Principle of Mathematical Induction, Fibonacci Number Prove that F1F2 + F2F3 + F3F4 + ...+ F2n - 1F2n = (F2n)^2.

Prove that (Fn+1)^2 - Fn Fn+2 = (- 1)^n

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Fill in the function table. when f(x) = x^2 , f(x) = 2x - 1 and f(x) = x^2 - 2x + 1 x f(x) -3 -2 -1 2 4.

Express 66 as a product of prime numbers.

Theory of Numbers (IX) Principle of Mathematical Induction Fibonacci Number Prove that

Theory of Numbers (VIII) Principle of Mathematical Induction Fibonacci Number Pro

Suppose that F1 = 1, F2 = 1, F3 =1, F4 = 3, F5 = 5, and in general Fn = Fn-1 + Fn-2 for n ≥ 3 ( Fn is called the nth Fibonacci number.) Prove that F1 + F2 + F3 +...+ Fn = F(n + 2) - 1

Meaning is given to the sum and product of all the natural numbers and then it is shown that: 1+2+3+4+... = -1/12 1*2*3*4*... = sqrt(2*pi). See the attached file.

Prove that 1^3 + 2^3 + 3^3 + ... + n^3 = (1 + 2 + 3 + ... + n)^2

Assume that n is odd and a is a primitive root mod n. Let b be an integer with b ≡ a(mod n) and gcd (b, 2n) =1. Show that b is a primitive root mod 2n.

Use Quadratic Congruences to evaluate the following fractions. See attached file for full problem description.

Quadratic Congruences. See attached file for full problem description. Let p be an odd prime. Complete the proof of the question "For which odd primes p is LS(2, p) = 1?" by showing that if and that if .

Prove that there are no integers x, y, and z such that x^2 + y^2 + z^2 = 999.

See attached file. This problem is from the Text Book Mathematical Methods in the Physical sciences 2nd Edition by Mary. L. Boas. Chapter 12. Section 5 problem number 6

List all irreducible polynomial of degree 2, 3 and 4 over F22juu2332. Prove your assertion. Please see the attached file for the fully formatted problems.

List each of the 15 elements x, x^2,...,x^15 of F2[x]/<x^4 +x + 1> as polynomials of degree at most three over F22.j.uu2332 Please see the attached file for the fully formatted problems.

Let P(z) and Q(z) be complex polynomials of degree m and n respectively, such that Q(z) has distinct roots z1...zn. Assume that n >= m+2 . Prove that P(z1)/Q'(z1) +...+ P(zn)/Q'(zn) (hint: evaluate ∫c P(z)/Q(z) dz for a suitable curve C).