Problem 5 - Elementary Algebra

Question

The product is \(-a^{2}+b^{2}\) and a factor is -1 . Find the other factor.

Step-by-Step Solution

Answer

Answer: The other factor is \(a^{2}-b^{2}\).

Step 1: Identify the product and the given factor

The product is given as \(-a^{2}+b^{2}\) and the given factor is -1.

Step 2: Divide the product by the given factor

Since one factor is given as -1, divide the given product by -1 to find the other factor: \[ \frac{-a^{2}+b^{2}}{-1} \]

Step 3: Simplify

By dividing each part of the expression by -1, we get the other factor as: \[ a^{2} - b^{2} \] The other factor of the product \(-a^{2}+b^{2}\) is \(a^{2}-b^{2}\).

Key Concepts

Factoring Expressions

Factoring expressions is a significant concept in algebra. It involves breaking down an algebraic expression into simpler components, called factors. These factors, when multiplied together, produce the original expression.

To factor an expression, we look for common elements or apply algebraic techniques. For example, consider the expression \(-a^{2} + b^{2}\). We know one factor is \(-1\), so we can divide the entire expression by \(-1\) to uncover the other factor.

This results in the expression \(a^{2} - b^{2}\), making \(a^{2} - b^{2}\) the other factor. Factoring expressions can make solving algebraic equations more manageable, allowing you to simplify and better understand complex problems.

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. A polynomial can be as simple as \(x + 1\) or as complex as \(-a^{2} + b^{2}\).

In this case, the given polynomial is \(-a^{2} + b^{2}\), which is of degree 2 because the highest power of the variable is 2. High-power polynomials can often be intimidating, but breaking them into smaller factors like we did with \(-a^{2} + b^{2}\) by factoring out \(-1\) can simplify them.

Understanding the nature of polynomials allows you to factor and solve them more effectively, helping to unlock a range of mathematical problems.

Negative Factors

Negative factors, such as \(-1\) in our example, play a crucial role in altering expressions. They change the sign of every term in the expression they factor out from. For example, factoring out \(-1\) from \(-a^{2} + b^{2}\) flips the signs, yielding \(a^{2} - b^{2}\). This sign change is pivotal in simplifying and rearranging algebraic expressions.

When dealing with negative factors, remember that they can impact the equation's terms significantly. Keep them in check to avoid any errors during your calculations.

Negative factors require careful attention because they transform the expression into a form that can be easier to manipulate, eventually solving equations in a more straightforward manner.