91Ó°ÊÓ

When dealing with multiplying polynomials, one of the first steps is to handle the coefficients. Coefficients are the numerical parts of your terms. In our example, the coefficients are \(-\frac{1}{3}\) and \(\frac{2}{5}\). To multiply these fractions, we'd perform the multiplication separately: multiply the numerators and then the denominators. So, it goes like this: \(-\frac{1}{3} \times \frac{2}{5}\). This becomes \(-\frac{1 \times 2}{3 \times 5} = -\frac{2}{15}\). It's crucial to keep track of the negative sign throughout as it affects the final outcome.

Always remember:

• Multiply the numerators together.
• Multiply the denominators together.
• Check the signs. A negative times a positive gives you a negative.

After that, your coefficient is ready to be combined with the other component of your polynomial term, the variable.

The properties of exponents come into play when you have the same base raised to two powers that need to be multiplied together. In this exercise, the base is \(y\), with exponents 2 and 1. According to the multiplication rule of exponents, \(a^m \times a^n = a^{m+n}\). So, for our example, \(y^2 \times y\) will become \(y^{2+1} = y^3\).

This rule is consistent for any variables:

• Add the exponents when the bases are the same during multiplication.
• This keeps any potential confusion at bay, especially when dealing with negative or fractional exponents.

Using these properties correctly helps streamline the process and ensures accuracy in your results.

Once the coefficients and variables have been handled separately, combine the results to achieve the simplified expression. From our steps, after multiplying and simplifying, the coefficient was \(-\frac{2}{15}\) and the variable part was \(y^{3}\). Together, the expression simplifies neatly into \(-\frac{2}{15}y^{3}\).

Simplifying means that the expression is expressed in the most efficient terms without losing its value. Some pointers to remember:

• Ensure you have performed all operations as required, like simplifying any fractions if possible.
• Ensure the variable part follows the properties of exponents correctly.

By doing this, we have a final solution that is neat and easily understandable.