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Chapter 5: Problem 7

Find each product. $$ 14 x^{2} y^{3}\left(-2 x^{5} y\right) $$

Short Answer

Expert verified
-28 x^{7} y^{4}

Step by step solution

01

- Multiply the coefficients

First, multiply the numerical coefficients together. Here, the coefficients are 14 and -2.Therefore, \(14 \times -2 = -28\).
02

- Multiply the variables with the same base

Next, multiply the variables with the same base by adding their exponents. For the variable \(x\): \(x^{2} \times x^{5} = x^{2+5} = x^{7}\).For the variable \(y\): \(y^{3} \times y = y^{3+1} = y^{4}\).
03

- Write the complete product

Combine the results from Step 1 and Step 2 to write the complete product. g value is: \(-28 x^{7} y^{4}\).

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Key Concepts

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

multiplying coefficients
When we talk about multiplying polynomials, the first step is to deal with their coefficients. Coefficients are the numerical parts of the terms you're working with. To multiply them, you simply perform the normal multiplication operation you would use for regular numbers.

In the given exercise, we have coefficients 14 and -2. Multiplying these gives: \(14 \times -2 = -28\).

This step is very straightforward. Always begin with the coefficients before moving on to the variables. It simplifies the process when you break it down into smaller, manageable steps.
adding exponents
The next core concept is dealing with exponents. When you multiply variables with the same base, you need to add their exponents. This comes from the properties of exponents in algebra.

For example, in the polynomial multiplication given, you have variables like \(x^{2} \text{ and } x^{5}\). To combine these, add the exponents: \(x^{2+5} = x^{7}\). Similarly, for variables like \(y^{3} \text{ and } y\): \(y^{3+1} = y^{4}\).

Make sure to always keep the bases the same when performing this operation. Adding exponents only applies when the variables are identical.
variables with same base
Understanding variables with the same base is crucial. It means being able to recognize and pair variables that can be combined by adding their exponents.

In our exercise, \(x^{2} \text{ and } x^{5}\) are variables with the same base (x), and \(y^{3} \text{ and } y\) are variables with the same base (y).

Recognizing this pattern allows you to apply the exponent addition rule effectively. This concept greatly simplifies polynomial multiplication and is an important shortcut in algebra.

Practice with more examples to become comfortable with identifying and working with variables of the same base.

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