(x-2-x-7)-5.jpeg "(x2-x-1)(x 2-x-7)-5")
Factoring the Expression (x²-x-1)(x²-x-7)-5
This article will guide you through factoring the expression (x²-x-1)(x²-x-7)-5. We'll explore different approaches and demonstrate how to find its simplified form.
Step 1: Simplifying the Expression
First, we can simplify the expression by expanding the product of the two quadratic factors:
(x²-x-1)(x²-x-7)-5 = x⁴ - 2x³ - 6x² + 8x + 7 - 5
Combining like terms, we get:
x⁴ - 2x³ - 6x² + 8x + 2
Step 2: Recognizing Potential Patterns
The expression now has a leading term (x⁴) and a constant term (2). We can try to factor this by looking for pairs of factors that multiply to 2 and add up to the coefficient of the x³ term (-2).
- Factors of 2 are (1, 2) and (-1, -2).
We can see that -1 and -2 add up to -3, not -2. This indicates that a simple factoring by grouping might not be straightforward.
Step 3: Using Substitution (Optional)
To make the expression appear simpler, we can substitute:
y = x² - x
This transforms our expression to:
(y - 1)(y - 7) - 5
Expanding this, we get:
y² - 8y + 7 - 5 = y² - 8y + 2
Step 4: Factoring the Quadratic
Now we have a simple quadratic expression in terms of 'y'. We can factor it as:
y² - 8y + 2 = (y - 4 + √14)(y - 4 - √14)
Step 5: Substituting Back
Finally, substitute back y = x² - x:
(x² - x - 4 + √14)(x² - x - 4 - √14)
Conclusion
The fully factored form of the expression (x²-x-1)(x²-x-7)-5 is (x² - x - 4 + √14)(x² - x - 4 - √14). Remember that this factoring process involves recognizing patterns, simplifying expressions, and understanding the properties of quadratics.