(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)+x^2

3 min read Jun 17, 2024

Factoring the Expression: (x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)+x^2

This article explores the process of factoring the expression (x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)+x^2. We'll demonstrate how to simplify this expression into a more compact and manageable form.

Expanding the Expression

First, we can expand the expression by multiplying out the individual terms:

(x-a)(x-b) = x^2 - ax - bx + ab
(x-b)(x-c) = x^2 - bx - cx + bc
(x-c)(x-a) = x^2 - cx - ax + ac

Combining these with the remaining term, our expression becomes:

x^2 - ax - bx + ab + x^2 - bx - cx + bc + x^2 - cx - ax + ac + x^2

Combining Like Terms

Next, we combine the like terms:

3x^2 - 2ax - 2bx - 2cx + ab + bc + ac

Factoring by Grouping

To further simplify, we can factor by grouping:

Group the terms with x^2: 3x^2
Group the terms with x: -2ax - 2bx - 2cx
Group the constant terms: ab + bc + ac

This gives us: 3x^2 + (-2ax - 2bx - 2cx) + (ab + bc + ac)

Now, factor out a -2x from the second group and an a from the third group:

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

Final Factored Form

Finally, we can rewrite the expression in a more compact form:

(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)+x^2 = 3x^2 - 2x(a + b + c) + a(b + c)

This is the factored form of the original expression.

Conclusion

By expanding the expression, combining like terms, and factoring by grouping, we successfully simplified the expression (x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)+x^2 into its factored form. This process demonstrates the importance of recognizing patterns and applying algebraic manipulation techniques to simplify complex expressions.