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(x-3)(x-1)

3 min read Jun 17, 2024
(x-3)(x-1)

Factoring and Expanding (x-3)(x-1)

This expression represents the product of two binomials: (x-3) and (x-1). Let's explore how to factor and expand it.

Expanding the Expression

To expand the expression, we use the distributive property (often referred to as FOIL - First, Outer, Inner, Last). Here's how it works:

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

Now, combine the terms:

(x-3)(x-1) = x² - x - 3x + 3

Simplify by combining like terms:

(x-3)(x-1) = x² - 4x + 3

Factoring the Expression

We can also factor the expression x² - 4x + 3 back into its original form (x-3)(x-1). Here's how:

  1. Find two numbers that add up to -4 (the coefficient of the x term) and multiply to 3 (the constant term). These numbers are -3 and -1.
  2. Rewrite the expression using these numbers: x² - 3x - x + 3
  3. Factor by grouping: (x² - 3x) + (-x + 3)
  4. Factor out the common factors: x(x-3) - 1(x-3)
  5. Factor out the common binomial: (x-3)(x-1)

Applications

The expression (x-3)(x-1) can be used in various mathematical contexts, including:

  • Solving quadratic equations: Setting the expanded form of the expression (x² - 4x + 3) equal to zero and solving for x will give us the roots of the quadratic equation.
  • Graphing quadratic functions: The expression can represent a quadratic function. By finding the roots and vertex, we can graph the function.
  • Algebraic manipulation: The factored form can be helpful in simplifying more complex algebraic expressions.

Understanding how to factor and expand this expression is a crucial skill in algebra and its applications.

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