(x^2-5x+4)(x^2-9)

Jasonbradley

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Jun 17, 2024

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Factoring and Expanding (x² - 5x + 4)(x² - 9)

This expression involves two quadratic expressions multiplied together. We can simplify this expression by:

1. Factoring each quadratic:

   • x² - 5x + 4 factors into (x - 4)(x - 1)
   • x² - 9 is a difference of squares and factors into (x + 3)(x - 3)

2. Multiplying the factored expressions:

   • (x - 4)(x - 1)(x + 3)(x - 3)

3. Expanding the product:

   • We can expand this product by systematically multiplying each term in the first expression by each term in the second expression. This can be done using the distributive property or by using a technique like the FOIL method.

Here's the expansion in detail:

(x - 4)(x - 1)(x + 3)(x - 3)

1. Expand (x - 4)(x - 1):

   • (x² - 5x + 4)(x + 3)(x - 3)

2. Expand (x + 3)(x - 3):

   • (x² - 5x + 4)(x² - 9)

3. Expand (x² - 5x + 4)(x² - 9):

   • x⁴ - 5x³ + 4x² - 9x² + 45x - 36

4. Combine like terms:

   • x⁴ - 5x³ - 5x² + 45x - 36

Therefore, the expanded and simplified form of (x² - 5x + 4)(x² - 9) is x⁴ - 5x³ - 5x² + 45x - 36.