(x-3)(x+2)(x+4)

3 min read Jun 17, 2024

Factoring and Expanding: A Look at (x-3)(x+2)(x+4)

This expression represents the product of three binomials: (x-3), (x+2), and (x+4). We can explore its properties in two ways: factoring and expanding.

Factoring

Factoring involves breaking down a polynomial into simpler expressions, usually binomials or monomials, that multiply together to produce the original polynomial. In this case, we already have the factored form of the expression.

Expanding

Expanding involves multiplying out the terms to obtain a single polynomial. To do this, we can use the distributive property repeatedly:

Start with the first two binomials: (x-3)(x+2) = x(x+2) - 3(x+2) = x² + 2x - 3x - 6 = x² - x - 6
Now multiply the result by the third binomial: (x² - x - 6)(x+4) = x²(x+4) - x(x+4) - 6(x+4) = x³ + 4x² - x² - 4x - 6x - 24 = x³ + 3x² - 10x - 24

Therefore, the expanded form of (x-3)(x+2)(x+4) is x³ + 3x² - 10x - 24.

Analyzing the Expanded Form

The expanded form reveals several important features:

Degree: The highest power of x is 3, making the expression a cubic polynomial.
Leading Coefficient: The coefficient of the term with the highest power of x is 1.
Constant Term: The constant term is -24, which is the product of the constant terms in the original binomials: -3 * 2 * 4 = -24.

Conclusion

By understanding both the factored and expanded forms of (x-3)(x+2)(x+4), we gain insight into its properties and can explore further applications in algebra and calculus.