(m – 1)(m2 + M + 1)(m + 4)(m – 4)

3 min read Jun 16, 2024

Factoring and Simplifying the Expression (m – 1)(m2 + m + 1)(m + 4)(m – 4)

This expression involves several factors, and we can simplify it by applying the concepts of difference of squares and sum of cubes.

(m - 1): This is a simple linear factor.
(m² + m + 1): This factor resembles the expansion of a cube: (a + b)³ = a³ + 3a²b + 3ab² + b³. However, it's not a perfect cube.
(m + 4) and (m - 4): These factors represent the difference of squares: a² - b² = (a + b)(a - b).

Difference of Squares: Let's start by simplifying the last two factors: (m + 4)(m - 4) = m² - 16
Sum of Cubes (for a specific case): While (m² + m + 1) is not a perfect cube, we can use a specific case of the sum of cubes formula: a³ + b³ = (a + b)(a² - ab + b²)

Notice that if we set a = m and b = 1, we get: m³ + 1³ = (m + 1)(m² - m + 1)

Since we have (m² - m + 1), we can rewrite this as: m³ + 1 = (m + 1)(m² - m + 1)
Combining the Results: Now we can combine all the simplified factors: (m – 1)(m² + m + 1)(m + 4)(m – 4) = (m - 1)(m² - m + 1)(m² - 16)

We can expand the last two factors, but it's not strictly necessary for simplification. The expression is now in a more compact and manageable form.

Final Simplified Form: (m - 1)(m² - m + 1)(m² - 16)

Important Note: This expression cannot be further factored into simpler linear factors using standard techniques.