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(m-2)(m-1)(m+3)

3 min read Jun 16, 2024
(m-2)(m-1)(m+3)

Factoring and Expanding (m-2)(m-1)(m+3)

This expression represents the product of three linear factors: (m-2), (m-1), and (m+3). We can approach it in two ways:

1. Expanding the Expression

We can expand the expression by multiplying the factors step by step:

  1. Multiply the first two factors: (m-2)(m-1) = m² - m - 2m + 2 = m² - 3m + 2

  2. Multiply the result by the third factor: (m² - 3m + 2)(m+3) = m³ + 3m² - 3m² - 9m + 2m + 6 = m³ - 7m + 6

Therefore, the expanded form of (m-2)(m-1)(m+3) is m³ - 7m + 6.

2. Finding the Roots

The expression (m-2)(m-1)(m+3) is equal to zero when any of the factors are equal to zero. This leads to the following solutions:

  • m - 2 = 0 => m = 2
  • m - 1 = 0 => m = 1
  • m + 3 = 0 => m = -3

Therefore, the roots of the equation (m-2)(m-1)(m+3) = 0 are m = 2, m = 1, and m = -3.

Applications

Understanding how to factor and expand expressions like (m-2)(m-1)(m+3) is crucial in various mathematical fields, including:

  • Algebra: Solving equations, simplifying expressions, and understanding the relationship between factors and roots.
  • Calculus: Finding derivatives and integrals of polynomial functions.
  • Linear Algebra: Working with matrices and vectors.

In addition to its mathematical significance, understanding the factorization and expansion of expressions like this can also be helpful in other areas such as:

  • Physics: Modeling physical phenomena using equations.
  • Engineering: Designing and analyzing structures and systems.
  • Computer Science: Developing algorithms and data structures.

Overall, understanding the relationship between factored and expanded forms of expressions provides a valuable tool for solving various mathematical problems and understanding the underlying principles of many scientific and engineering fields.

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