(a+b)c^3-(a^2+ab+b^2)c^2+a^2b^2

4 min read Jun 16, 2024

Factoring the Expression (a+b)c^3-(a^2+ab+b^2)c^2+a^2b^2

This article will explore the factoring of the algebraic expression: (a+b)c^3-(a^2+ab+b^2)c^2+a^2b^2

We can approach this problem using several techniques, but we will focus on a method involving grouping and recognizing patterns.

Step 1: Grouping Terms

Begin by grouping the terms with similar factors:

(a+b)c^3 - (a^2+ab+b^2)c^2 + a^2b^2 = [(a+b)c^3 - (a^2+ab+b^2)c^2] + a^2b^2

Step 2: Factoring out Common Factors

Next, factor out the common factors from each group:

From the first group: c^2(a+b)c - c^2(a^2+ab+b^2) = c^2[(a+b)c - (a^2+ab+b^2)]
From the second group: a^2b^2 remains unchanged.

Now the expression becomes: c^2[(a+b)c - (a^2+ab+b^2)] + a^2b^2

Step 3: Recognizing the Pattern

Notice that the expression inside the square brackets, (a+b)c - (a^2+ab+b^2), resembles a difference of squares pattern.

We can rewrite it as:

(a+b)c - (a^2+ab+b^2) = (a+b)c - (a+b)(a+b) = (a+b)[c - (a+b)]

Step 4: Final Factoring

Substituting this back into our expression, we get:

c^2[(a+b)[c - (a+b)]] + a^2b^2 = c^2(a+b)(c-a-b) + a^2b^2

Finally, we can factor out a common factor of (a+b):

c^2(a+b)(c-a-b) + a^2b^2 = (a+b)[c^2(c-a-b) + ab^2]

Therefore, the fully factored form of the expression is: (a+b)[c^2(c-a-b) + ab^2].

Conclusion

By using grouping, factoring out common terms, and recognizing the difference of squares pattern, we successfully factored the expression (a+b)c^3-(a^2+ab+b^2)c^2+a^2b^2 into (a+b)[c^2(c-a-b) + ab^2]. This process demonstrates the importance of pattern recognition and strategic manipulation in simplifying algebraic expressions.