(x-1)(x-2)(x+7)(x+8)+8

5 min read Jun 17, 2024

Factoring and Exploring the Expression (x-1)(x-2)(x+7)(x+8)+8

This article will explore the fascinating expression (x-1)(x-2)(x+7)(x+8)+8. We'll break down its factorization, analyze its roots, and discuss its key features.

Factoring the Expression

At first glance, the expression might seem daunting. However, with a little manipulation, we can reveal its underlying structure and simplify it.

Grouping: Let's group the first two terms and the last two terms:
```
[(x-1)(x-2)] [(x+7)(x+8)] + 8
```
Expanding: Expand the grouped terms:
```
(x² - 3x + 2)(x² + 15x + 56) + 8
```
Substitution: To make things easier, let's introduce a substitution. Let y = x² + 6x. Notice that:
- x² - 3x + 2 = (x² + 6x) - 9x + 2 = y - 9x + 2
- x² + 15x + 56 = (x² + 6x) + 9x + 56 = y + 9x + 56
Substituting these back into the expression:
```
(y - 9x + 2)(y + 9x + 56) + 8
```
Expanding and Simplifying: Now we can expand and simplify:
```
y² + 47y + 112 + 8 = y² + 47y + 120
```
Factoring the Quadratic: Finally, we can factor the quadratic in terms of y:
```
(y + 3)(y + 40)
```
Substituting Back: Substituting back for y, we get the factored form:
```
(x² + 6x + 3)(x² + 6x + 40)
```

Analyzing the Expression's Roots

The roots of the expression are the values of x that make the expression equal to zero. To find the roots, we need to solve the following equations:

x² + 6x + 3 = 0
x² + 6x + 40 = 0

These equations are quadratic equations. We can use the quadratic formula to find their solutions:

x = (-b ± √(b² - 4ac)) / 2a

For the first equation, a = 1, b = 6, and c = 3. For the second equation, a = 1, b = 6, and c = 40. Plugging these values into the quadratic formula, we can find the four roots of the expression.

Key Features

Degree: The expression is a quartic polynomial, meaning it has a degree of 4.
Symmetry: Due to the way the expression is factored, it exhibits symmetry around the line x = -3. This can be seen in the graph of the expression.
Real Roots: The expression has four real roots, which are the x-values where the graph intersects the x-axis.
Graph: The graph of the expression is a curve with four x-intercepts corresponding to the roots. It has a general shape similar to a "W".

Applications

This expression, though seemingly complex, can be applied in various mathematical contexts:

Calculus: The expression can be used to find the maxima and minima of functions.
Algebra: Its factorization provides a deeper understanding of the expression's behavior and its relationship to other expressions.
Geometry: The expression can be used to model certain geometric shapes and relationships.

Conclusion

The expression (x-1)(x-2)(x+7)(x+8)+8, although initially appearing complex, can be simplified and analyzed using various mathematical techniques. Understanding its factorization and its roots reveals its intricate structure and potential applications in different fields.