(x-10)(x-10)

Jasonbradley

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Jun 17, 2024

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Understanding (x - 10)(x - 10)

The expression (x - 10)(x - 10) is a product of two binomials, both of which are identical. This means it represents a special case of multiplication that has a specific pattern and can be simplified.

Expanding the Expression

To understand what (x - 10)(x - 10) represents, we need to expand it. This involves using the distributive property of multiplication.

1. Distribute the first term (x) of the first binomial:

   • x * (x - 10) = x² - 10x

2. Distribute the second term (-10) of the first binomial:

   • -10 * (x - 10) = -10x + 100

3. Combine the results from steps 1 and 2:

   • x² - 10x - 10x + 100

4. Simplify by combining like terms:

   • x² - 20x + 100

Recognizing the Pattern: Perfect Square Trinomial

The expanded form, x² - 20x + 100, is a perfect square trinomial. This means it follows a specific pattern:

• The first term (x²) is the square of the first term of the binomial (x).
• The last term (100) is the square of the second term of the binomial (10).
• The middle term (-20x) is twice the product of the first and second terms of the binomial (2 * x * -10 = -20x).

Using the Pattern for Simplification

Knowing this pattern allows us to directly simplify expressions like (x - 10)(x - 10) without going through the full expansion process:

1. Square the first term: x²
2. Multiply the two terms and double the result: 2 * x * -10 = -20x
3. Square the second term: (-10)² = 100
4. Combine the terms: x² - 20x + 100

This pattern helps us quickly and efficiently simplify expressions involving identical binomials.