Problem 637 Simplify each expression. $$(n... [FREE SOLUTION]

91影视

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 637

Simplify each expression. $$(n-8)\left(n^{2}-4 n+11\right)$$

Short Answer

Expert verified

$n^{3} - 12n^2 + 43n - 88$

Step by step solution

- Distribute the First Term

Multiply the first term in the first polynomial, which is $n$, by each term in the second polynomial: $n^{2}$, $-4n$, and $11$.

- Distribute the Second Term

Multiply the second term in the first polynomial, which is $-8$, by each term in the second polynomial: $n^{2}$, $-4n$, and $11$.

- Combine Like Terms

Add all the like terms together to simplify the expression. Like terms are the terms that have the same variable raised to the same power.

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 property

The distributive property is fundamental when handling polynomial multiplication. This principle allows us to multiply each term in one polynomial by each term in the other polynomial. In our exercise, we apply the distributive property twice.
First, we distribute the polynomial $(n-8)$ with $(n^2 - 4n + 11)$. We need to multiply each term inside the brackets one by one:

For the first step, distribute the first term:

\text{Distribute $n$:} \ n \times n^2 = n^3 \ n \times (-4n) = -4n^2 \ n \times 11 = 11n

Second, distribute the second term:
\ (-8) \times n^2 = -8n^2 \ (-8) \times (-4n) = 32n \ (-8) \times 11 = -88

After distributing all terms, we collect the resulting polynomial terms for the next steps.

combining like terms

Combining like terms is the process of summing or subtracting terms that have the same variables raised to identical powers. Following our distribution, we should have the following terms:

\ n^3 - 4n^2 + 11n - 8n^2 + 32n - 88

Group the like terms together:
The $n^3$ term stands alone since there are no other $n^3$ terms:
$n^3$

For the $-4n^2$ and $-8n^2$, we combine them as follows: \ -4n^2 - 8n^2 = -12n^2, resulting in one $n^2$ term:
-12n^2

Similarly, combine the linear $11n + 32n$ terms: \ 11n + 32n = 43n, giving us one $n$ term:
43n

Lastly, we account for the constant, which is $-88$. After grouping and combining the like terms, we reduce the polynomial to its simplified form.

simplifying expressions

Simplifying an expression means reducing it to its most concise form. This process eliminates unnecessary terms and combines similar entities. From our previous steps, we simplified the polynomial to:

\ n^3 - 12n^2 + 43n - 88
This final polynomial cannot be simplified further because:

\text{All terms have distinct degrees (powers of $n$):} \ The $n^3$ term stands alone, representing the highest power of $n$.

\text{The $n^2$ term has no other matching powers, making $-12n^2$ distinct.}
\text{The linear term $43n$ is uniquely linear, and}

\text{The constant $-88$ stands on its own.}

By following these steps, we can systematically simplify polynomial expressions, making them easier to understand and work with in various mathematical contexts. The key is to always distribute fully, combine like terms accurately, and ensure the expression is in its simplest form.

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

Simplify each expression. $$(n-8)\left(n^{2}-4 n+11\right)$$

Short Answer

Step by step solution

- Distribute the First Term

- Distribute the Second Term

- Combine Like Terms

Key Concepts

distributive property

combining like terms

simplifying expressions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Theoretical and Mathematical Physics

Geometry

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.