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Chapter 4: Problem 8

Find each product. $$ 10 p^{2}\left(5 p^{3}\right) $$

Short Answer

Expert verified
50 p^{5}

Step by step solution

01

- Understand the Exercise

The problem asks to find the product of two terms: \(10 p^{2}\) and \(5 p^{3}\).
02

- Multiply the Coefficients

Multiply the numerical coefficients (numbers) first. In this case, multiply 10 by 5: \[ 10 \times 5 = 50 \]
03

- Multiply the Powers of p

Next, multiply the powers of \(p\) by adding their exponents. The exponents are 2 and 3. Add them together: \[ p^{2} \times p^{3} = p^{2+3} = p^{5} \]
04

- Combine the Results

Combine the result from Step 2 and Step 3 to get the final product: \[ 50 p^{5} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coefficients
In the given exercise, coefficients are the numerical parts of the terms. For the expression \(10 p^{2}(5 p^{3})\), the coefficients are 10 and 5. When multiplying polynomials, always start by multiplying these numbers together.

To multiply the coefficients: \[ 10 \times 5 = 50 \] It's important to handle this step separately from the variables to keep things clear and simple.

Remember, coefficients are just like the plain numbers we're used to working with. Keep them aside, multiply them directly, and then move on to the variables.
Exponents
Once you've handled the coefficients, the next step involves dealing with the exponents. In the polynomials \(10 p^{2}(5 p^{3})\), the exponents are 2 and 3, associated with the variable \(p\).

When multiplying variables, you add their exponents together. So for \[ p^{2} \times p^{3} = p^{2+3} = p^{5} \] this shows the rule applied.

Always remember: Adding exponents is an essential step when dealing with like variables in multiplication. This rule helps simplify seemingly complex expressions and is crucial for solving polynomial multiplication problems accurately.
Product of Terms
The final step in our exercise is to combine the results. We've already multiplied the coefficients (10 and 5) to get 50, and we've added the exponents (2 and 3) to get 5. Now, we combine these two results.

The product of the terms is: \[ 50 p^{5} \] By following a systematic approach鈥攎ultiplying coefficients first, then adding the exponents鈥攜ou can simplify any polynomial multiplication.

This method ensures that you keep track of all parts of the terms, making the exercise easier to handle and understand. Practicing this will aid in mastering the multiplication of polynomials efficiently.

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