Math Problem Statement

factorise a ^ 3 + 1 / a ^ 3 + 2a + 2 / a

Solution

To factorize the expression a3+1a3+2a+2aa^3 + \frac{1}{a^3} + 2a + \frac{2}{a}, let's break it down step by step.

Step 1: Group the terms

The expression can be grouped as follows: (a3+1a3)+(2a+2a)\left(a^3 + \frac{1}{a^3}\right) + \left(2a + \frac{2}{a}\right)

Step 2: Factorize each group

Notice that each group resembles a form that can be factored using algebraic identities.

  • The first group a3+1a3a^3 + \frac{1}{a^3} can be recognized as a part of the identity for the sum of cubes. Using the identity:

a3+1a3=(a+1a)(a21+1a2)a^3 + \frac{1}{a^3} = \left(a + \frac{1}{a}\right) \left(a^2 - 1 + \frac{1}{a^2}\right)

But this identity simplifies better when combined with the second group.

  • The second group 2a+2a2a + \frac{2}{a} can be factored as:

2(a+1a)2\left(a + \frac{1}{a}\right)

Step 3: Combine and factor further

Now, combine the factorizations from both groups:

(a+1a)(a21+1a2)+2(a+1a)\left(a + \frac{1}{a}\right) \left(a^2 - 1 + \frac{1}{a^2}\right) + 2\left(a + \frac{1}{a}\right)

We can factor out the common term (a+1a)\left(a + \frac{1}{a}\right):

(a+1a)[(a21+1a2)+2]\left(a + \frac{1}{a}\right)\left[\left(a^2 - 1 + \frac{1}{a^2}\right) + 2\right]

Simplify the expression inside the brackets:

(a+1a)(a2+1a2+1)\left(a + \frac{1}{a}\right)\left(a^2 + \frac{1}{a^2} + 1\right)

Final Factorized Form:

Thus, the factorized form of the expression is:

(a+1a)(a2+1a2+1)\boxed{\left(a + \frac{1}{a}\right)\left(a^2 + \frac{1}{a^2} + 1\right)}

Would you like more details or have any questions?

Here are five related questions you might find helpful:

  1. How do you factorize expressions involving the difference of cubes?
  2. What are the key algebraic identities used in factorization?
  3. How do you simplify complex fractions like 1a3\frac{1}{a^3}?
  4. Can you factorize a4+1a4+2a2+2a2a^4 + \frac{1}{a^4} + 2a^2 + \frac{2}{a^2} in a similar way?
  5. What is the importance of recognizing patterns in algebraic expressions?

Tip: Always check for common factors across terms before applying more complex factorization techniques.

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Math Problem Analysis

Mathematical Concepts

Algebra
Factorization
Algebraic Identities

Formulas

Sum of cubes identity
Factorization of expressions involving fractions

Theorems

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Suitable Grade Level

Advanced High School