91Ó°ÊÓ

Factoring polynomials is a crucial skill in algebra that allows us to simplify expressions and solve equations more efficiently. A polynomial is an expression containing variables and constants, combined using addition, subtraction, and multiplication. To factor a polynomial, we aim to express it as a product of smaller polynomials.

• In this particular problem, we start by observing the terms in the equation: \( n^3 + 49n = 0 \).
• Notice that both terms in this expression share a common factor, which is the variable \( n \).

By factoring out the common factor \( n \), we simplify the equation to \( n(n^2 + 49) = 0 \). This transformation is key because it allows us to apply the zero-product property, making it easier to solve the equation. Factoring polynomials like this often requires recognizing patterns or shared factors among the terms, and it sets the foundation for solving more complex polynomial equations.

Once we have factored a polynomial, solving the corresponding equation becomes straightforward. The zero-product property tells us that if a product of factors equals zero, then at least one of the factors must be zero. This principle helps us solve the equation \( n(n^2 + 49) = 0 \).
First, we set each factor equal to zero:

• \( n = 0 \) leads directly to one solution for \( n \).
• For \( n^2 + 49 = 0 \), setting this factor to zero helps us explore other potential solutions.

The straightforward part comes from the factor \( n = 0 \), which immediately gives us one solution. The challenge often lies in solving the other factor \( n^2 + 49 = 0 \). Understanding the need to solve each factor separately is critical in using the zero-product property effectively.

In the realm of real numbers, the equation \( n^2 + 49 = 0 \) might initially seem unsolvable, but this is where complex numbers come into play. Complex numbers extend our number system by introducing \( i \), the imaginary unit, where \( i^2 = -1 \).
To solve \( n^2 + 49 = 0 \), we rearrange it to \( n^2 = -49 \). The square root of a negative number isn't possible in the real numbers. However, with complex numbers, the solution becomes feasible:

• We take the square root of both sides, getting \( n = \pm \sqrt{-49} \).
• This simplifies further to \( n = \pm 7i \).

Thus, in addition to the real solution \( n=0 \), the complex solutions are \( n = 7i \) and \( n = -7i \). Understanding complex numbers enriches our ability to find solutions to equations that don't fit within the real number framework. This expands our mathematical toolkit, allowing us to solve a broader range of problems.