Practice Abstract Algebra Problems

Introduction to practice abstract algebra problems:

In algebra, we use a, b,c.... x, y and z to denote numbers. Performing addition, subtraction, multiplication, and division of roots on this symbols and real numbers, we obtain what are called algebraic expressions

The following are some examples for algebraic equations:

2x + 3 = x + 6,

Symbols in an algebraic expression are called  variables in Algebra

In Algebra  the word can be understood by a simple example. In the equation x + 5 = 9, the left hand side is the addition (sum) of two parts x and 5. If we add  (–5) to each side of the equation, we get

(x + 5) + (–5) = 9 + (–5) or x + [5 + (–5)] = 9 – 5 or x + 0 = 4 or x = 4.

Here 9 and −5 are reunited to get 4. It is said to be  algebra.  

Practice Problems in Algebra

Abstract algebra problems using Subtract 2x^3 – 3x^2 + 1 from x^3 + 5x^2 – 4x – 6 for practice.

Solution:  Associative and Distributive properties, we have

(x^3 + 5x^2 – 4x – 6) – (2x^3 – 3x^2 + 1)

= x^3 + 5x^2 – 4x – 6 – 2x^3 + 3x^2 - 1

= x^3 – 2x^3 + 5x^2 + 3x^2 – 4x – 6 - 1

= (x^3 – 2x^3) + (5x^2 + 3x^2) + (–4x) + (– 6 - 1)

= –x^3 + 8x^2 – 4x – 7.

Abstract algebra problems using find the product of x^3 – 2x^2 + 4 and 2x^2 + 3x – 1 .

Solution: (x^3 – 2x^2 +4) (2x^2 + 3x – 1)

= x^3 (2x^2 + 3x – 1) + (– 2x^2) (2x^2 + 3x – 1) + (+ 4) (2x^2 + 3x – 1)

= (2x5 + 3x4 – x^3) + (– 4x4 – 6x^3 + 2x^2) + (+ 8x^2 + 12x - 4)

= 2x5 + 3x4 – x^3 – 4x4 – 6x^3 + 2x^2 + 8x^2 + 12x - 4

= 2x5 + (3x4 – 4x4) + (–x^3 – 6x^3) + (2x^2 + 8x^2) + (+ 12x) -4

= 2x5 – x4 – 7x^3 + 10x^2 + 12x - 4.

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Problem Using the Identities for Practice:

Abstract algebra problems using Identities (a + b + c)2 – 2(ab + bc + ca) = a2 + b2 + c2 for practice.

Solution:

Abstract algebra problems (i) (2x + y + 2z)2

(2x + y + 2z)2 = [(2x) + y + (2z)]2 = (2x)2 + y2 + (2z)2 + 2(2x)y + 2y(2z) + 2(2z)(2x)

= 4x^2 + y2 + 4z2 + 4xy + 4yz + 8zx.

Abstract algebra problems (ii) (x – 2y + z)2

(x – 2y + z)2 = [x + (–2y) + z]2= x^2 + (–2y)2 + z2 + 2x(–2y) +2 (–2y)z + 2zx

= x^2 + 4y2 + z2 – 4xy – 4yz + 2zx.

Abstract algebra problems (iii) (2p – 3q – r)2

(2p – 3q – r)2 = [(2p) + (–3q) + (–r)]2

= (2p)2 + (–3q)2 + (–r)2 + 2(2p) (–3q) + 2(–3q) (–r) + 2(–r)(2p).

= 4p2 + 9q2 + r2 – 12pq + 6qr – 4rp.

Abstract algebra problems (iv) (2a + 3b − 2c)2

= (2a + 3b – 2c)2 = [(2a) + (3b) + (–2c)]2

= (2a)2 + (3b)2 + (–2c)2 + 2(2a)(3b) + 2(3b)(–2c) + 2(–2c)(2a)

= 4a2 + 9b2 + 4c2 + 12ab – 12bc – 8ca.