Problem 56 Multiply. $$\left(5 y^{2}+8 y-2\... [FREE SOLUTION]

Chapter 9: Problem 56

Multiply. $$\left(5 y^{2}+8 y-2\right)(3 y-8)$$

Short Answer

Expert verified

The result of the multiplication is $15y^3 - 16y^2 - 70y + 16$.

Step by step solution

- Apply Distributive Law to the First Term

Apply the distributive law of multiplication over addition to the first term of the first binomial. This means multiplying the first term, $5y^2$ of $(5y^2 + 8y - 2)$ with each term in the second binomial $(3y - 8)$. This gives us $(15y^3 - 40y^2)$.

- Apply Distributive Law to the Second Term

Repeat the process for the second term of the first binomial, $8y$. Multiply it with each term in the second binomial, giving $(24y^2 - 64y)$.

- Apply Distributive Law to the Third Term

Finally, repeat this process with the last term in the first binomial, $-2$, resulting in $-6y + 16$.

- Combine Like Terms

Combine all the obtained expressions from the previous steps. This leaves us with the polynomial $15y^3 - 40y^2 + 24y^2 - 64y - 6y + 16$. Combine like terms to simplify this expression to $15y^3 - 16y^2 - 70y + 16$.

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.

Distributive Law

The Distributive Law is a foundational principle in algebra that allows us to multiply a single term across a bracketed expression. When faced with a polynomial multiplication task like $(5y^2 + 8y - 2)(3y - 8)$,
we apply the distributive law to break down the process step by step.For each term in the first polynomial, multiply it by every term in the second polynomial.

For $5y^2$, multiply it with $3y$ and $-8$ to get $15y^3$ and $-40y^2$.
Next, multiply $8y$ by each term in $3y - 8$, obtaining $24y^2$ and $-64y$.
For the final term, $-2$, multiply it across $3y - 8$ to get $-6y$ and $+16$.

The distributive law turns a complex equation into smaller, manageable pieces.

Combining Like Terms

Once you have applied the distributive law, you'll end up with a list of terms:$15y^3 - 40y^2 + 24y^2 - 64y - 6y + 16$.
Each of these terms might have the same variable part, known as "like terms."The goal in this step is to add or subtract the coefficients of the like terms.

Identify terms with the same variable degree, like $y^2$ or $y$.
Combine $-40y^2$ and $24y^2$ to get $-16y^2$.
For the $y$ terms, combine $-64y$ and $-6y$ to obtain $-70y$.

This process reduces redundancy and makes the polynomial easier to interpret.

Expression Simplification

Expression Simplification is the final step that consolidates all previous work into a neat, concise format. After combining like terms, you get:$15y^3 - 16y^2 - 70y + 16$.
Simplifying means writing the polynomial in its most straightforward form possible, arranged by degree of terms, usually starting with the highest.

Notice how this expression now has fewer terms than what we started with.
The expression is ordered from the highest degree $y^3$ to the constant.

Simplification ensures clarity, allowing you to easily see the behavior and characteristics of the polynomial.

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Multiply. $$\left(5 y^{2}+8 y-2\right)(3 y-8)$$

Short Answer

Step by step solution

- Apply Distributive Law to the First Term

- Apply Distributive Law to the Second Term

- Apply Distributive Law to the Third Term

- Combine Like Terms

Key Concepts

Distributive Law

Combining Like Terms

Expression Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Logic and Functions

Calculus

Theoretical and Mathematical Physics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.