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(x-10)(x-2)

2 min read Jun 17, 2024
(x-10)(x-2)

Factoring and Expanding (x-10)(x-2)

This expression represents the product of two binomials: (x-10) and (x-2). We can analyze it in two ways:

Expanding the Expression

To expand the expression, we use the FOIL method (First, Outer, Inner, Last):

  1. First: Multiply the first terms of each binomial: x * x = x²
  2. Outer: Multiply the outer terms of the binomials: x * -2 = -2x
  3. Inner: Multiply the inner terms of the binomials: -10 * x = -10x
  4. Last: Multiply the last terms of each binomial: -10 * -2 = 20

Now, combine the terms:

x² - 2x - 10x + 20 = x² - 12x + 20

Therefore, the expanded form of (x-10)(x-2) is x² - 12x + 20.

Factoring the Expression

We can also factor the expression x² - 12x + 20 back into its original form:

  1. Find two numbers that multiply to 20 and add up to -12. These numbers are -10 and -2.
  2. Rewrite the middle term (-12x) using these numbers: x² - 10x - 2x + 20
  3. Factor by grouping: (x² - 10x) + (-2x + 20) x(x - 10) - 2(x - 10)
  4. Factor out the common binomial: (x - 10)(x - 2)

This confirms that the factored form of x² - 12x + 20 is (x-10)(x-2).

Applications

Understanding how to expand and factor binomials is crucial in algebra, especially when working with quadratic equations. It allows us to manipulate expressions and solve for unknown variables.

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