(x-1)(x-3)(x-5)(x-7)+15 因数分解

2 min read Jun 17, 2024
(x-1)(x-3)(x-5)(x-7)+15 因数分解

Factoring the expression (x-1)(x-3)(x-5)(x-7)+15

This problem involves a clever trick to factor a seemingly complex expression. Let's break it down step-by-step:

1. Recognizing the pattern:

Observe that the first part of the expression consists of four consecutive odd numbers subtracted from x. This suggests a potential pattern we can exploit.

2. Rearranging terms:

Let's rearrange the expression by grouping the first and last terms, and the middle two terms:

[(x-1)(x-7)] * [(x-3)(x-5)] + 15

3. Expanding and simplifying:

Expanding the grouped terms, we get:

(x² - 8x + 7) * (x² - 8x + 15) + 15

Notice that both expressions within the parentheses share the same first two terms (x² - 8x). This is crucial for the next step.

4. Substitution for simplification:

Let's substitute y for (x² - 8x):

(y + 7) * (y + 15) + 15

5. Expanding and factoring:

Expanding the expression:

y² + 22y + 105 + 15 = y² + 22y + 120

Factoring the quadratic:

(y + 10)(y + 12)

6. Substituting back:

Now, we substitute back (x² - 8x) for y:

(x² - 8x + 10)(x² - 8x + 12)

7. Final factorization:

Finally, we factor the remaining quadratics:

(x - 2)(x - 6)(x - 4)(x - 3)

Therefore, the fully factored expression is (x - 2)(x - 6)(x - 4)(x - 3).

This method demonstrates how recognizing patterns and strategic substitutions can lead to a simplified solution for seemingly complicated expressions.

Related Post


Featured Posts