91Ó°ÊÓ

The distributive property helps us expand expressions and multiply binomials efficiently. When you hear the term 'distributive property,' think about distributing or spreading out one operation over another. In our example, we will distribute each term in the first binomial to each term in the second binomial.

For \((3x + 2)(3x + 5)\), we distribute each term as follows:

- Multiply \(3x\) by \(3x\)
- Multiply \(3x\) by \(5\)
- Multiply \(2\) by \(3x\)
- Multiply \(2\) by \(5\)

This gives us multiple smaller multiplication problems to solve. This step ensures every term in the first binomial is appropriately multiplied by every term in the second binomial.

After using the distributive property, we often end up with multiple terms that need to be added together. In this step, we focus on combining like terms.

Like terms are terms that have the same variable raised to the same power. For instance, \(15x\) and \(6x\) are like terms because both have \(x\) raised to the first power.

In the context of our problem, after multiplying the terms, we get: \(9x^2\), \(15x\), \(6x\), and \(10\). We can see that \(15x\) and \(6x\) are like terms.
- Adding \(15x\) and \(6x\) gives us \(21x\).

Thus, our next step is combining these like terms to simplify the expression further.

Polynomial multiplication might sound complex, but it's simply about applying the distributive property and combining like terms.

Here's a step-by-step on how to perform polynomial multiplication:

- Write down the binomials: \((3x + 2)(3x + 5)\).
- Distribute each term from the first binomial to each term in the second binomial, giving: \(3x \times 3x + 3x \times 5 + 2 \times 3x + 2 \times 5\).
- Multiply the individual terms: \(9x^2\), \(15x\), \(6x\), and \(10\).
- Combine the like terms: \(15x\) and \(6x\) to get \(21x\).

This means the final expression after multiplying and simplifying is \(9x^2 + 21x + 10\).

Polynomial multiplication is a fundamental skill in algebra. Practicing these steps will make it easier to handle more complex problems in the future.