/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 24 Find each product. $$ -7 y\l... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 24

Find each product. $$ -7 y\left(3+5 y^{2}-2 y^{3}\right) $$

Short Answer

Expert verified
-21y - 35y^{3} + 14y^{4}

Step by step solution

01

- Distribute the term

To find the product, distribute \(-7y\) to each term inside the parentheses. This means you will multiply \(-7y\) with each of the terms \(3\), \(5 y^{2}\), and \(-2 y^{3}\).
02

- Multiply \-7y\ by 3

Start with the first term inside the parentheses: \(-7y \times 3 = -21y\).
03

- Multiply \-7y\ by \5 y^{2}\

Next, multiply \(-7y\) by \(5 y^{2}\): \(-7y \times 5 y^{2} = -35 y^{3}\).
04

- Multiply \-7y\ by \-2 y^{3}\

Finally, multiply \(-7y\) by \(-2 y^{3}\): \(-7y \times -2 y^{3} = 14 y^{4}\).
05

- Combine all terms

Now, put all the terms together: \(-21y - 35y^{3} + 14y^{4}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Polynomial multiplication
When we talk about polynomial multiplication, we are referring to multiplying a polynomial by another polynomial or a monomial. In the given example, we are multiplying \( -7y \) by a polynomial \( 3 + 5y^{2} - 2y^{3} \). To do this, we use the distributive property to expand and simplify the expression.

Just remember, polynomial multiplication can involve:
  • Distributing each term of one polynomial to every term of the other polynomial
  • Using properties of exponents when terms involve variables raised to powers
By following these guidelines, polynomial multiplication becomes much more manageable.
Distributive property
The distributive property is a fundamental concept in algebra that helps us simplify expressions, particularly when dealing with multiplication over addition or subtraction. It states that for any terms a, b, and c:
\[ a(b + c) = ab + ac \]

In our example, \( -7y \) is distributed to each term inside the parentheses. We use the distributive property to multiply \( -7y \) by \( 3, 5y^{2}, \) and \( -2y^{3} \):
  • \( -7y \times 3 = -21y \)
  • \( -7y \times 5y^{2} = -35y^{3} \)
  • \( -7y \times -2y^{3} = 14y^{4} \)
This property simplifies the process, ensuring we correctly multiply each term without missing any parts of the expression.
Combining like terms
Combining like terms is essential for simplifying polynomials. Like terms are terms that have the same variable raised to the same power. When you combine like terms, you add or subtract their coefficients, making the expression simpler.

In our example, after using the distributive property, we ended up with:
\[ -21y - 35y^{3} + 14y^{4} \]

Since there are no additional like terms to combine, the expression remains as it is. However, in problems where there are like terms, always:
  • Identify terms with the same variable and power
  • Add or subtract their coefficients
Following these steps ensures your final polynomial is in its simplest form.

Remember, combining like terms is crucial in making polynomials more manageable and easier to work with!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ 3 p^{3}\left(2 p^{2}+5 p\right)\left(p^{3}+2 p+1\right) $$

Add or subtract as indicated. \((8 a b+2 a-3 b)-(6 a b-2 a+3 b)\)

Without actually performing the operations, mentally determine the coefficient of the \(x\) -term in the simplified form of $$\left(-8 x^{2}-3 x+2\right)-\left(4 x^{2}-3 x+8\right)-\left(-2 x^{2}-x+7\right)$$

Pollux, one of the brightest stars in the night sky, is 33.7 light-years from Earth. If one light-year is about \(6,000,000,000,000 \mathrm{mi}\) (that is, 6 trillion mi), about how many miles is Pollux from Earth? (Data from The World Almanac and Book of Facts.)

Graph each equation by completing the table of values. $$ \begin{aligned} &y=(x-4)^{2}\\\ &\begin{array}{|l|l|l|l|l|l} x & 2 & 3 & 4 & 5 & 6 \\ \hline y & & & & & \end{array} \end{aligned} $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.