(x-1)(x+1)(x+2)(x+4)-72

4 min read Jun 17, 2024

Factoring the Expression (x-1)(x+1)(x+2)(x+4) - 72

This article will explore how to factor the expression (x-1)(x+1)(x+2)(x+4) - 72. We'll utilize various algebraic techniques to simplify and arrive at the factored form.

Recognizing the Pattern

Observe that the first part of the expression, (x-1)(x+1)(x+2)(x+4), consists of four factors, which are pairs of binomials that differ only in their signs. This suggests a specific pattern:

Difference of Squares: (x-1)(x+1) can be factored as x² - 1² = x² - 1.
Grouping: We can group (x+2)(x+4) to facilitate further factoring.

Applying the Pattern

Factor the difference of squares: (x-1)(x+1)(x+2)(x+4) - 72 = (x² - 1)(x+2)(x+4) - 72
Expand the product of the grouped binomials: (x² - 1)(x+2)(x+4) - 72 = (x² - 1)(x² + 6x + 8) - 72
Multiply the remaining expressions: (x² - 1)(x² + 6x + 8) - 72 = x⁴ + 6x³ + 7x² + 6x + 8 - 72
Simplify by combining constants: x⁴ + 6x³ + 7x² + 6x + 8 - 72 = x⁴ + 6x³ + 7x² + 6x - 64

Factoring the Simplified Expression

The simplified expression, x⁴ + 6x³ + 7x² + 6x - 64, can be factored using various techniques, such as the Rational Root Theorem or by trial and error.

Here's one approach:

Find a possible root: We can observe that x=1 is a root of the expression.
Use the Factor Theorem: Since x=1 is a root, (x-1) is a factor of the expression. We can use polynomial long division or synthetic division to find the remaining factor:

x⁴ + 6x³ + 7x² + 6x - 64 ÷ (x-1) = x³ + 7x² + 14x + 20
Factor the remaining cubic expression: The cubic expression x³ + 7x² + 14x + 20 can be factored by grouping:

x³ + 7x² + 14x + 20 = (x³ + 7x²) + (14x + 20) = x²(x+7) + 2(7x + 10)
Factor out the common binomial: Notice that the terms (x+7) appear in both expressions. We can factor this out:

x²(x+7) + 2(7x + 10) = (x+7)(x² + 2)

Final Factored Form

Therefore, the factored form of the original expression is:

(x-1)(x+1)(x+2)(x+4) - 72 = (x-1)(x+7)(x² + 2)