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Chapter 4: Problem 10

Find each product. $$ 5 y(-6 y+1) $$

Short Answer

Expert verified
-30y^2 + 5y

Step by step solution

01

- Apply the Distributive Property

To find the product of the expression, apply the distributive property: multiply each term inside the parentheses by the term outside the parentheses. This means multiplying 5y by -6y and then multiplying 5y by 1.
02

- Multiply 5y by -6y

Multiply the coefficients (5 and -6) to get -30 and then multiply the variables: y * y = y^2. So, 5y * -6y = -30y^2.
03

- Multiply 5y by 1

Multiply 5y by 1, which simply gives 5y.
04

- Combine the Terms

Combine the results from steps 2 and 3: -30y^2 + 5y.

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Key Concepts

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

distributive property
The distributive property is a key concept in polynomial multiplication. It helps you break down more complex expressions into simpler parts that are easier to handle. When you apply the distributive property, you multiply each term inside the parentheses by the term outside the parentheses.

For instance, in the exercise, we have:
  • Given expression: 5y(-6y + 1)
  • Apply the distributive property: 5y * (-6y) and 5y * 1

So, each term inside the parentheses (-6y and 1) gets multiplied by 5y, the term outside. This allows us to separate the problem into smaller, manageable pieces.

Remember to always distribute each term fully to avoid any mistakes in calculations.
combining like terms
Combining like terms is essential when you're simplifying polynomial expressions. Like terms are terms that have the same variable parts raised to the same power.

After using the distributive property in the exercise, we get two terms:
  • -30y^2 (from multiplying 5y by -6y)
  • 5y (from multiplying 5y by 1)

These terms are different because one has a y^2 and the other has a y. Since they are not like terms, we cannot combine them. Instead, we write the final answer as a sum of these two terms: -30y^2 + 5y.

Always pay attention to the exponents when combining like terms. Only terms with the exact same variable parts can be combined.
coefficient multiplication
Coefficient multiplication refers to multiplying the numerical parts (coefficients) of terms separately from their variable parts.

For example, in the exercise, we need to multiply 5y by -6y. Let's break it down:
  • First, multiply the coefficients: 5 * -6 = -30
  • Then multiply the variables: y * y = y^2

Putting these together, we get:
  • 5y * -6y = -30y^2

Coefficient multiplication helps manage parts of the problem separately, making it simpler to execute the steps accurately. Always multiply the numbers first and then handle the variables.

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Most popular questions from this chapter

Solve each problem. In \(2016,\) the national debt of the U.S. government was about \(\$ 19.32\) trillion. The U.S. population at that time was approximately 323.1 million. (Data from U.S. Census Bureau, U.S. Department of the Treasury.) (a) Write the population using scientific notation. (b) Write the amount of the national debt using scientific notation. (c) Using the answers for parts (a) and (b), calculate how much debt this is per American. Write this amount in standard notation to the nearest dollar.

Write each number in standard notation. $$ 5.92 \times 10^{0} $$

Write each number in scientific notation. $$ 0.830 $$

Use the FOIL method to find each product. $$ (z-w)(3 z+4 w) $$

The perimeter \(x\) of a square with sides of length \(s\) is given by the formula \(x=4 s\). (a) Solve for \(s\) in terms of \(x\). (b) If \(y\) represents the area of this square, write \(y\) as a function of the perimeter \(x\) (c) Use the composite function of part (b) to find the area of a square with perimeter 6 .
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