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Chapter 10: Problem 546

Multiply. $$-5 y\left(125 y^{3}-1\right)$$

Short Answer

Expert verified
-625 y^{4} + 5 y.

Step by step solution

01

Distribute the Latex placed correctly

Distribute -5 y to each term inside the parenthesis. This means you will multiply -5 y with 125 y^{3} and -5 y with -1 separately.
02

Multiply Latex placed correctly

First, multiply -5 y and 125 y^{3}: -5 y 脳 125 y^{3} = -5 脳 125 脳 y^{1+3}. Simplify the numbers and exponents: -5 脳 125 = -625 and y^{1+3} = y^{4}. Therefore, the result is -625 y^{4}.
03

Multiply the second pair

Next, multiply -5 y and -1: -5 y 脳 -1 = 5 y.
04

Combine the results

Now, combine the results from Steps 2 and 3: -625 y^{4} + 5 y.

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Key Concepts

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

Multiplying Polynomials
Multiplying polynomials involves distributing each term in one polynomial to every term in the other polynomial. Begin by multiplying the coefficients (the numerical parts) together. Then, apply the exponent rules for the variables.

For example, in the problem \(-5y(125y^3 - 1)\), we distribute \(-5y\) to each term inside the parenthesis. This means \(-5y \cdot 125y^3 + (-5y \cdot -1)\).

Key steps are:
  • Distribute each term separately.
  • Multiply the coefficients.
  • Add exponents for variables of the same base.
Following these steps simplifies the process of multiplying polynomials.
Negative Coefficients
Dealing with negative coefficients can initially seem tricky, but it's quite straightforward. When you multiply a negative coefficient with a positive one, the result is negative. When both coefficients are negative, the result is positive.

For instance, in the exercise \(-5y \cdot 125y^3\), \(-5 \cdot 125 = -625\). This is because a negative times a positive gives a negative result. Similarly, with \(-5y \cdot -1 = 5y\), since a negative times a negative is positive.

Remember these basic rules to manage negative coefficients:
  • Negative \cdot Positive = Negative
  • Negative \cdot Negative = Positive
Consistently applying these rules will help you avoid sign errors in your calculations.
Combining Like Terms
Combining like terms involves simplifying an expression by summing coefficients of terms with the same variable and exponent. Like terms have the same variable raised to the same power.

In our example, after distributing and multiplying, we get \(-625y^4 + 5y\). These terms cannot be combined further because \(-625y^4\) and \(+5y\) have different exponents (4 and 1, respectively).

Steps to combine like terms:
  • Identify like terms (same variable and exponent).
  • Add or subtract the coefficients of these terms while keeping the variable part unchanged.
  • Simplify the expression as much as possible.
Combining like terms helps streamline expressions and makes them easier to work with.

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