(x-2)(x%2B3)(x%2B4)-84.jpeg "(x-1)(x-2)(x+3)(x+4)-84")
Factoring and Solving the Expression (x-1)(x-2)(x+3)(x+4)-84
This article explores the process of factoring and solving the expression (x-1)(x-2)(x+3)(x+4)-84. We will utilize algebraic manipulation and factoring techniques to simplify the expression and find its roots.
1. Recognizing the Pattern
Observe that the first four terms of the expression resemble a product of four binomials. This suggests a potential pattern that can be exploited for factoring.
2. Rearranging and Grouping
Let's rearrange the terms and group them strategically:
(x-1)(x+3)(x-2)(x+4) - 84
We can now see a more apparent pattern: the product of two pairs of binomials.
3. Expanding the Groups
Expand each group of binomials:
[(x² + 2x - 3)][(x² + 2x - 8)] - 84
Notice that both expressions within the brackets share the same quadratic term, x² + 2x.
4. Substitution for Simplification
To make the expression more manageable, let's substitute y = x² + 2x:
(y - 3)(y - 8) - 84
5. Expanding and Factoring
Expand the product and simplify:
y² - 11y + 24 - 84
= y² - 11y - 60
Now, we can factor the quadratic:
(y - 15)(y + 4)
6. Resubstituting and Solving
Substitute back y = x² + 2x:
(x² + 2x - 15)(x² + 2x + 4)
Now we have two quadratic expressions. We can factor each one further:
[(x + 5)(x - 3)][(x² + 2x + 4)]
The second quadratic expression (x² + 2x + 4) doesn't factor further using real numbers.
Therefore, the factored form of the original expression is:
(x + 5)(x - 3)(x² + 2x + 4)
7. Finding the Roots
To find the roots, we need to solve for x when the expression equals zero:
(x + 5)(x - 3)(x² + 2x + 4) = 0
This gives us three possible solutions:
- x + 5 = 0 => x = -5
- x - 3 = 0 => x = 3
- x² + 2x + 4 = 0 (This quadratic has no real roots)
Therefore, the real roots of the expression are x = -5 and x = 3.
Conclusion
By carefully rearranging, grouping, and applying substitution, we successfully factored the expression (x-1)(x-2)(x+3)(x+4)-84. This process revealed the real roots of the expression, which are x = -5 and x = 3. This exercise demonstrates how strategic manipulation and factoring techniques can simplify complex expressions and lead to valuable insights.