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Chapter 5: Problem 14

Add $$ \left(5 n^{3}-n^{2}+4 n-3\right)+\left(2 n^{3}-4 n^{2}+n+3\right) $$

Short Answer

Expert verified
7n^3 - 5n^2 + 5n

Step by step solution

01

- Write down the given polynomials

First, write down the two polynomials that need to be added: ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewlineewline ewline} ewlineewline ewline (5n^3 - n^2 + 4n - 3) + (2n^3 - 4n^2 + n + 3) ewlineWrite each polynomial in a single expression to make the next step easier. ewline
02

- Identify and combine like terms

Combine the coefficients of the terms with the same powers of n: ewline ewline ewline ewline ewline ewlineewline ewlineCombine like terms as follows: ewline(5n^3 + 2n^3) + (-n^2 - 4n^2) + (4n + n) + (-3 + 3) ewline
03

- Simplify the expression

Add the coefficients of the like terms to obtain the simplified polynomial: ewline ewline ewlineCombine like terms to get the simplified polynomial: ewline(5 + 2)n^3 + (-1 - 4)n^2 + (4 + 1)n + (-3 + 3) ewlineWhich simplifies to: ewline7n^3 - 5n^2 + 5n ewlineThe constant terms -3 and 3 cancel each other out. ewline

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Key Concepts

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

Like Terms
In polynomial addition, it's important to identify and combine like terms. Like terms are terms that have the same variable raised to the same power. For example, in the expression \(5n^3 + 2n^3\), both terms are like because they both contain \(n^3\).

To add these terms together, you simply add their coefficients (the numbers in front of the variables). Here, the coefficients are 5 and 2, so \(5n^3 + 2n^3 = 7n^3\).

This process applies to all terms in the polynomial. It's essential to match the variables and their exponents to ensure you're working with like terms.
Coefficients
Coefficients are the numerical parts of the terms in a polynomial. They tell you how many units of each term you have. For example, in the term \(5n^3\), the coefficient is 5.

When adding polynomials, you focus on combining the coefficients of like terms. For instance, in the expression \( (5n^3 - n^2 + 4n - 3) + (2n^3 - 4n^2 + n + 3) \), you'll add the coefficients of the \(n^3\) terms first:

\(5n^3 + 2n^3\) which gives \(7n^3\).

Next, for the \(n^2\) terms, you combine \(-n^2\) and \(-4n^2\) to get \(-5n^2\).

The same process is used for the remaining terms. This method helps you simplify the polynomial efficiently.
Simplification
Simplification involves combining like terms to reduce a polynomial to its simplest form. This means grouping all terms with the same variables and exponents and summing their coefficients.

Let's take the expression \((5n^3 - n^2 + 4n - 3) + (2n^3 - 4n^2 + n + 3)\). First, write down the like terms together:

\((5n^3 + 2n^3) - (n^2 + 4n^2) + (4n + n) - (3 - 3)\).

Now, add the coefficients of the like terms:

1. \(5n^3 + 2n^3 = 7n^3\)
2. \(-n^2 - 4n^2 = -5n^2\)
3. \(4n + n = 5n\)
4. \(-3 + 3 = 0\)

So, the simplified polynomial is \(7n^3 - 5n^2 + 5n\). Simplification makes it easier to understand and work with the polynomial.

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Most popular questions from this chapter

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