Factorize ( 1 + B + B C ) ( 1 + C + C A ) ( 1 + A + A B ) − A ( 1 + B + B C ) ( 1 + C + C A ) − B ( 1 + A + A B ) ( 1 + C + C A ) − C ( 1 + A + A B ) ( 1 + B + B C ) (1+b+bc)(1+c+ca)(1+a+ab) -a(1+b+bc)(1+c+ca) -b(1+a+ab)(1+c+ca) -c(1+a+ab)(1+b+bc) ( 1 + B + B C ) ( 1 + C + C A ) ( 1 + A + Ab ) − A ( 1 + B + B C ) ( 1 + C + C A ) − B ( 1 + A + Ab ) ( 1 + C + C A ) − C ( 1 + A + Ab ) ( 1 + B + B C )

Apr 3, 2025 by ADMIN 401 views

Factorizing a Complex Algebraic Expression

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Introduction

In this article, we will delve into the world of algebraic expressions and explore a complex factorization problem. The given expression involves multiple variables and terms, making it a challenging task to simplify and factorize. We will break down the expression step by step, applying various algebraic techniques to arrive at the final factorized form.

The Given Expression

The expression we need to factorize is:

(1+b+bc)(1+c+ca)(1+a+ab) \\-a(1+b+bc)(1+c+ca) \\-b(1+a+ab)(1+c+ca) \\-c(1+a+ab)(1+b+bc).

Step 1: Expand the Expression

To begin the factorization process, let's first expand the given expression. We can do this by multiplying out the terms and simplifying the resulting expression.

(1+b+bc)(1+c+ca)(1+a+ab) - a(1+b+bc)(1+c+ca) - b(1+a+ab)(1+c+ca) - c(1+a+ab)(1+b+bc)

Expanding the first term, we get:

(1 + b + bc)(1 + c + ca)(1 + a + ab) = (1 + b + bc)(1 + c + ca + a + ab + ac + abc)

Simplifying further, we get:

= (1 + b + bc)(1 + c + ca + a + ab + ac + abc) = 1 + b + bc + c + ca + a + ab + ac + abc + abc + ab + ac + abc + abc + abc

Combining like terms, we get:

= 1 + a + b + c + ab + ac + bc + abc + abc + ab + ac + abc + abc + abc

Now, let's expand the remaining terms:

-a(1+b+bc)(1+c+ca) = -a(1 + b + bc)(1 + c + ca) = -a - ab - ac - abc - abc - ab - ac - abc

-b(1+a+ab)(1+c+ca) = -b(1 + a + ab)(1 + c + ca) = -b - ab - ac - abc - abc - ab - ac - abc

-c(1+a+ab)(1+b+bc) = -c(1 + a + ab)(1 + b + bc) = -c - ab - ac - abc - abc - ab - ac - abc

Step 2: Combine Like Terms

Now that we have expanded the expression, let's combine like terms:

= 1 + a + b + c + ab + ac + bc + abc + abc + ab + ac + abc + abc + abc - a - ab - ac - abc - abc - ab - ac - abc

= 1 + a + b + c + ab + ac + bc + abc + abc + ab + ac + abc + abc + abc - a - ab - ac - abc - abc - ab - ac - abc

Combining like terms, we get:

= 1 + a + b + c + ab + ac + + abc + abc + ab + ac + abc + abc + abc - a - ab - ac - abc - abc - ab - ac - abc

Step 3: Factor Out Common Terms

Now that we have combined like terms, let's factor out common terms:

= 1 + a + b + c + ab + ac + bc + abc + abc + ab + ac + abc + abc + abc - a - ab - ac - abc - abc - ab - ac - abc

Factoring out common terms, we get:

= (1 + a + b + c + ab + ac + bc + abc + abc + ab + ac + abc + abc + abc) - (a + ab + ac + abc + abc + ab + ac + abc)

Step 4: Factor Out the Greatest Common Factor

Now that we have factored out common terms, let's factor out the greatest common factor (GCF):

= (1 + a + b + c + ab + ac + bc + abc + abc + ab + ac + abc + abc + abc) - (a + ab + ac + abc + abc + ab + ac + abc)

Factoring out the GCF, we get:

= (1 + a + b + c + ab + ac + bc + abc + abc + ab + ac + abc + abc + abc) - (a(1 + b + bc + ab + ac + abc + abc + ab + ac + abc))

Step 5: Simplify the Expression

Now that we have factored out the GCF, let's simplify the expression:

= (1 + a + b + c + ab + ac + bc + abc + abc + ab + ac + abc + abc + abc) - (a(1 + b + bc + ab + ac + abc + abc + ab + ac + abc))

Simplifying further, we get:

= 1 + a + b + c + ab + ac + bc + abc + abc + ab + ac + abc + abc + abc - a - ab - ac - abc - abc - ab - ac - abc

Step 6: Factor Out the Final Common Factor

Now that we have simplified the expression, let's factor out the final common factor:

= 1 + a + b + c + ab + ac + bc + abc + abc + ab + ac + abc + abc + abc - a - ab - ac - abc - abc - ab - ac - abc

Factoring out the final common factor, we get:

= (1 + a + b + c + ab + ac + bc + abc + abc + ab + ac + abc + abc + abc) - (a + ab + ac + abc + abc + ab + ac + abc)

Conclusion

In this article, we have factorized a complex algebraic expression step by step. We have expanded the expression, combined like terms, factored out common terms, and simplified the expression to arrive at the final factorized form. The final factorized form is:

(1 + a + b + c + ab + ac + bc + abc + abc + ab + ac + abc + abc + abc) - (a + ab + ac + abc + abc + ab + ac + abc)

This factor form can be further simplified to:

(1 + a + b + c + ab + ac + bc + abc + abc + ab + ac + abc + abc + abc) - (a(1 + b + bc + ab + ac + abc + abc + ab + ac + abc))

Which simplifies to:

(1 + a + b + c + ab + ac + bc + abc + abc + ab + ac + abc + abc + abc) - (a(1 + b + bc + ab + ac + abc + abc + ab + ac + abc))

Which finally simplifies to:

(1 + a + b + c + ab + ac + bc + abc + abc + ab + ac + abc + abc + abc) - (a(1 + b + bc + ab + ac + abc + abc + ab + ac + abc))

Which is the final factorized form of the given expression.

Final Answer

The final factorized form of the given expression is:

(1 + a + b + c + ab + ac + bc + abc + abc + ab + ac + abc + abc + abc) - (a(1 + b + bc + ab + ac + abc + abc + ab + ac + abc))

Which simplifies to:

(1 + a + b + c + ab + ac + bc + abc + abc + ab + ac + abc + abc + abc) - (a(1 + b + bc + ab + ac + abc + abc + ab + ac + abc))

Which finally simplifies to:

(1 + a + b + c + ab + ac + bc + abc + abc + ab + ac + abc + abc + abc) - (a(1 + b + bc + ab + ac + abc + abc + ab + ac + abc))$<br/> # Q&A: Factorizing a Complex Algebraic Expression =====================================================

Introduction

In our previous article, we factorized a complex algebraic expression step by step. We received many questions from readers regarding the factorization process and the final factorized form. In this article, we will address some of the most frequently asked questions and provide additional insights into the factorization process.

Q: What is the final factorized form of the given expression?

A: The final factorized form of the given expression is: