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(x-3)(x+2) (x-6)(x-7)

3 min read Jun 17, 2024
(x-3)(x+2) (x-6)(x-7)

Factoring and Solving the Equation (x-3)(x+2)(x-6)(x-7) = 0

This expression represents a polynomial equation in factored form. Let's explore how to solve this equation and what it tells us about the polynomial.

Understanding the Factored Form

The equation is already factored, which makes it very easy to find the solutions (also called roots or zeros). Each factor represents a linear expression that equals zero when:

  • (x - 3) = 0 => x = 3
  • (x + 2) = 0 => x = -2
  • (x - 6) = 0 => x = 6
  • (x - 7) = 0 => x = 7

Therefore, the solutions to the equation (x-3)(x+2)(x-6)(x-7) = 0 are x = 3, x = -2, x = 6, and x = 7.

Graphical Interpretation

This equation represents a polynomial function of degree 4. The solutions we found correspond to the x-intercepts of the graph of this function.

  • The graph will cross the x-axis at the points (3, 0), (-2, 0), (6, 0), and (7, 0).

Expanding the Equation

While the factored form is useful for finding solutions, we can also expand the expression to get the standard polynomial form:

  1. Multiply the first two factors: (x-3)(x+2) = x² - x - 6

  2. Multiply the last two factors: (x-6)(x-7) = x² - 13x + 42

  3. Multiply the results from steps 1 and 2: (x² - x - 6)(x² - 13x + 42) = x⁴ - 14x³ + 59x² - 90x - 252

Therefore, the expanded form of the equation is x⁴ - 14x³ + 59x² - 90x - 252 = 0.

Note: While we can now easily solve the equation in factored form, expanding it is sometimes necessary to analyze the polynomial's properties further.

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