In the factor ring Z[i]/ < 3 + i> the element 12 - 5i+ < 3 +i> is equivalent to 2+ < 3 + i > O 7+ < 3+ i > O 5+ < 3 + i > O 1+ < 3 + i > O

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

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Q: Let z be a root of the polynomial X + 3X + 1. Write the inverse of z? – z + 1 in the form az? + bz +…

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Q: Find the multiplicative inverse of 3 in GF(13) domain using Fermat’s Little theorem.

A: We have to find the multiplicative inverse of 3 in GF(13) domain using Fermat's Little Theorem.…

Q: Show that the square of any integer is of the form 3m or 3m+1 but not of form 3m+2.

A: Proof :-

Q: a) Use Lagrange interpolation to find the unique quadratic polynomial f that passes through the…

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Q: Which of the following polynomials is irreducible over Q? * O x^4+x+1 3x 3-6x 2+x-2 O x*2+5x-6 ONone…

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Q: Use the Lagrange interpolation formula to show that if a polynomial in Pn has n + 1 zeros, then it…

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Q: Write the polynomial z - 2r3 -1 as product of distinct poly- nomials of degree 1 and 2.

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Q: Solve the polynomial congruence: x³ - 3x² +27= 0 (mod 225).

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Q: EXERCISE 8 1. DON'T SIMPLIFY the following. ONLY INDICATE whether or not the result of the product…

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Q: Identify the end behavior for the following polynomial a) ↑↑ 个个 8r7 +5x³2x² + 1 此 b) ↓ c) ↑↓ d) ↓↑

A: Given function fx=8x7+5x5-2x2+1 Identify the end behaviour of the polynomial.

Q: Determine which of the polynomials below is (are) irreducibleover Q.

A: According to our guidelines we can answer only one question and rest can be reposted.

Q: The generally polynomials are of the form Select one: anx+an-1x²-1 + ax + ao O b. an" + an-x-1+...+…

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Q: ) Use the distributive law to determine which answer represents the given expression as a polynomial…

A: Distributive law a(b + c) = ab + ac

Q: • Create two different types of problems about polynomial addition • Create two different types of…

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Q: how the element 2+√3 has a multiplicative inverse in Q(√3). Show that Q(√3) is a subfield of C

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Q: Which of the following polynomials is an element in the span of the polynomials 1 + x + x³ and x+x²?…

A: Definition:

Q: Show that if x is an integer, then gcd(x + 1, x2 – x + 1) divides 3. -

A: First, we find gcdx+1, x2-x+1 Now, factors of x+1 are x+1, 1 x2-x+1 is a quadratic equation of the…

Q: 29. The area of the shaded region shown below can be represented as (2x + 6) (2x + 6) - (r(x + 3)²).…

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Q: Please use method from Abstract Algebra, Show that the polynomial 37a² + 30a³ + a⁴ = 1500 does not…

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

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Q: Find the polynomial with thats has degree 3 and Zeros 2,-1 and 3

A: To find the polynomial with thats has degree 3 and Zeros 2, -1 and 3.

Q: Choose the polynomial that is written in standard form. A7x + 8x8+2 7x -3x¹y + 2x²y² + 5y³ 10xy +…

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Q: Use the definition of O-notation to prove that 15x3 + 8x + 4 is O(x3)

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Q: 3. Find the greatest common divisor of: (a) The polynomials r³ + x² + x and r³ +x* +x² + x in Z2.

A: The given polynomials are: x3+x2+x, x5+x4+x2+x The given group is ℤ2.

Q: Simplify the following expressions in the indicated polynomial rings: A. (3x³ + 2x² + 4x + 2) + (2x¹…

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Q: Indicate whether the polynomial is written in standard form or not. A) Standard form B) NOT standard…

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Q: Show that x19 + x8 + 1 has multiple zeros in some extension of Z3.

A: To Show :x19+x8+1 has multiple zeroes in some extension of Z3.

Q: 1) Name the polynomial a) 3x? + 9x - 1 b) 2x* - 9x? - 1

A: a) 3x2+9x-1The above polynomial has a degree 2⇒It is quadratic polynomial

Q: Form the polynomial function that has the following real ZEROS: -1, 1, 0, –3 Section 3.1 O a+3a + a²…

A: See attached file for a step by step explanation to find the required polynomial.

Q: Give all possible rational zeros for the following polynomial. 42) P(x) = 3x3 + 32x2 + 32x + 27 A)…

A: We have to find the all possible zeroes of the given polynomial.

Polynomials

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