(x-7)(x-5)(x-4)(x-2)-72

3 min read Jun 17, 2024

Factoring the Expression (x-7)(x-5)(x-4)(x-2)-72

This article will guide you through the process of factoring the expression (x-7)(x-5)(x-4)(x-2)-72.

Understanding the Expression

The expression represents a polynomial with four factors: (x-7), (x-5), (x-4), and (x-2), and a constant term of -72. Our goal is to simplify this expression by factoring it into a more manageable form.

The Strategy

We can factor this expression by using a combination of grouping and algebraic manipulation.

Rearrange the terms: Let's rearrange the expression to group the factors that share common terms:
```
[(x-7)(x-2)][(x-5)(x-4)] - 72
```
Expand the grouped terms: Expand each pair of factors:
```
(x² - 9x + 14)(x² - 9x + 20) - 72
```
Substitute for simplification: To make the expression easier to manipulate, let's substitute y = x² - 9x. This gives us:
```
(y + 14)(y + 20) - 72
```
Expand and simplify: Expand the product and simplify the expression:
```
y² + 34y + 280 - 72 
= y² + 34y + 208
```
Factor the quadratic expression: Now we have a quadratic expression in terms of 'y'. Factor this expression:
```
(y + 26)(y + 8)
```
Substitute back: Substitute back the original expression for 'y':
```
(x² - 9x + 26)(x² - 9x + 8)
```
Factor the remaining quadratics: Factor the two remaining quadratic expressions:
```
(x-2)(x-13)(x-1)(x-8)
```

Final Result

The factored form of the expression (x-7)(x-5)(x-4)(x-2)-72 is: (x-2)(x-13)(x-1)(x-8).

Summary

By using a combination of grouping, substitution, and factoring, we were able to factor the given expression into its simplest form. This process demonstrates how to approach complex factoring problems by breaking them down into smaller, more manageable steps.