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Chapter 5: Problem 10

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

Short Answer

Expert verified
-30y^2 + 5y

Step by step solution

01

Distribute the term

Multiply the term outside the parenthesis (5y) by each term inside the parenthesis (-6y and 1).
02

Multiply 5y by -6y

Calculate the product of 5y and -6y: \(5y \times -6y = -30y^2\).
03

Multiply 5y by 1

Calculate the product of 5y and 1: \(5y \times 1 = 5y\).
04

Combine the results

Add 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 fundamental concept in mathematics. It allows you to multiply a single term by each term within a set of parentheses. In this problem, we apply the distributive property to the expression \(5y(-6y+1)\).

Follow these steps to use the distributive property:
  • Take the term outside the parentheses, which is \(5y\).
  • Multiply it by each term inside the parentheses one by one.

So for the given expression, you would first multiply \(5y\) by \(-6y\), and then \(5y\) by \(1\). This method simplifies complex expressions and is essential for polynomial multiplication.
Polynomial Multiplication
In polynomial multiplication, we use the distributive property to multiply each term. Here, the terms are multiplied as follows:

  • First, multiply \(5y\) by \(-6y\), which gives \(5y \times -6y = -30y^2\).
  • Second, multiply \(5y\) by \(1\), resulting in \(5y \times 1 = 5y\).

When multiplying terms, remember to multiply both the coefficients (numbers) and the variables. For instance, in \(5y \times -6y\), multiply the coefficients (5 and -6) and the variables \(y\) and \(y\). The product of \(y \times y\) is \(y^2\). This gives you \(-30y^2\). Knowing and applying these steps rigorously helps you solve polynomial multiplication problems correctly.
Combining Like Terms
The final step in simplifying the expression \(5y(-6y+1)\) is combining like terms.

Like terms are terms that have the same variable raised to the same power. In the expression \(-30y^2 + 5y\), there are no like terms to combine because \(-30y^2\) and \(5y\) are different terms. One is a term with \(y^2\) and the other with \(y\).

Here鈥檚 a quick tip to identify like terms:
  • Check if the variables in the terms are the same.
  • Make sure the variables have the same exponent.
  • Combine only the numerical coefficients of like terms.

This process ensures that you present your final simplified expression correctly.

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