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Chapter 6: Problem 4

Multiply. $$\left(3 z^{7}\right)\left(5 z^{3}\right)$$

Short Answer

Expert verified
The multiplication of \(\left(3 z^{7}\right)\left(5 z^{3}\right)\) is equal to \(15z^{10}\).

Step by step solution

01

Multiply the coefficients

First, we need to multiply the coefficients (the numbers in front of the variables). In this case, we have the coefficients 3 and 5: \(3 \times 5 = 15\)
02

Combine the variables with their exponents

Next, we need to combine the variables with their exponents. Since both variables are "z" and they have the exponents 7 and 3, we can simply add the exponents: \(z^{7} \times z^{3} = z^{7+3}\)
03

Complete the multiplication

Finally, we will put together the result from step 1 and step 2: \(15z^{7+3} = 15z^{10}\) So the multiplication of \(\left(3 z^{7}\right)\left(5 z^{3}\right)\) is equal to \(15z^{10}\).

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Key Concepts

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

Multiplying Coefficients
In algebra, when we're dealing with multiplication of expressions like \( (3z^7)(5z^3) \), we first focus on the numerical part, known as the coefficients. These numbers are multiplied just like you would with regular numbers. Consider the coefficients from the exercise: **3** and **5**. To multiply them, you follow the fundamental rule of multiplication, which is repeatedly adding the number to itself. However, in multiplication, you only need to do it once:
  • 3 times 5 gives you 15, which is just 3 four more times or 5 three more times.
So, multiplying the coefficients, we get **15**. Simple, right? Don鈥檛 overthink it! Getting this first step right is crucial in polynomial multiplication as it forms the base of our final result.
Combining Exponents
When multiplying terms with the same base, in this case, "z," we apply the rule of exponents. This principle states that you should keep the base and add the exponents.Looking at our example, we have \( z^7 \) and \( z^3 \). Both have the same base "z."
  • To combine these terms, add the exponents:
    - \( z^{7+3} = z^{10} \).
This process of adding exponents makes sense because it reflects the idea of multiplying powers. With \( z^7 \) and \( z^3 \), you essentially have 7 \( z's \) being multiplied by another 3 \( z's \), totaling 10 \( z's \) together. So, remember, when the bases are the same, just add the exponents.
Polynomial Multiplication
Polynomial multiplication involves several steps, beginning with multiplying coefficients and combining exponents, then synthesizing these results into a single expression.In our problem, we multiplied the coefficients (3 and 5) to get 15 and added the exponents (7 and 3) to arrive at \( z^{10} \).Combine these two results to complete the polynomial multiplication:
  • The final product of \((3z^7)(5z^3)\) is
    - \( 15z^{10} \).
By understanding each step separately鈥攎ultiplying the coefficients and combining the exponents鈥攜ou'll efficiently solve any polynomial multiplication problem. It鈥檚 merely about putting these straightforward techniques together to form one coherent solution.

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