(x-7)(x-1)

Jasonbradley

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Jun 17, 2024

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Factoring and Expanding (x - 7)(x - 1)

This expression represents the product of two binomials: (x - 7) and (x - 1). We can explore its properties through factoring and expanding:

Factoring

(x - 7)(x - 1) is already in its factored form. This means it's expressed as the product of two or more simpler expressions.

Expanding

To expand the expression, we use the distributive property (also known as FOIL - First, Outer, Inner, Last):

1. First: Multiply the first terms of each binomial: x * x = x²
2. Outer: Multiply the outer terms: x * -1 = -x
3. Inner: Multiply the inner terms: -7 * x = -7x
4. Last: Multiply the last terms: -7 * -1 = +7

Now, combine the terms:

x² - x - 7x + 7

Simplify by combining the like terms:

x² - 8x + 7

Therefore, the expanded form of (x - 7)(x - 1) is x² - 8x + 7.

Key Points

• Factored Form: (x - 7)(x - 1)
• Expanded Form: x² - 8x + 7
• Zero Product Property: The factored form helps us find the solutions (roots) of the quadratic equation x² - 8x + 7 = 0. The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we can set each factor equal to zero and solve for x:
  • x - 7 = 0 => x = 7
  • x - 1 = 0 => x = 1

These solutions represent the x-intercepts of the parabola defined by the quadratic equation.