Question

Simplify and write the resulting polynomial in descending order of degree. $$-3 n^{2}-2 n+7-4 n^{2}$$

Short answer

Question: Simplify the polynomial $$-3 n^2 -2 n + 7 -4 n^2$$ and write it in descending order of degree. Answer: The simplified polynomial is $$-7 n^2 -2 n + 7$$.

Step-by-step solution

Identify like terms

In the given polynomial, $$-3 n^2$$ and $$-4 n^2$$ are like terms, because they both have the same variable (n) raised to the same power (2). Similarly, $$-2 n$$ is a like term of itself.

Combine like terms

Now that we have identified the like terms, we can combine them: $$(-3 n^2) + (-4 n^2) = (-3 - 4) n^2 = -7 n^2$$ We can now rewrite the polynomial with the combined terms: $$-7 n^2 -2 n + 7$$

Write the polynomial in descending order of degree

The polynomial is now simplified, and we just need to ensure it is written in descending order of degree. The highest degree term is $$-7 n^2$$, followed by the first-degree term $$-2 n$$, and lastly, the constant $$7$$. Therefore, the final polynomial in descending order of degree is: $$-7 n^2 -2 n + 7$$

Key concepts

Combining Like Terms

When simplifying polynomials, one of the first steps is to combine like terms. Like terms are terms that have exactly the same variable components raised to the same power. For instance, in the polynomial \( -3 n^2 - 2 n + 7 - 4 n^2 \) the terms \( -3 n^2 \) and \( -4 n^2 \) are like terms, since they both include the variable \( n \) raised to the second power.

To combine these like terms, simply add or subtract their coefficients (the numbers in front of the variables). In this case, \( -3 \) and \( -4 \) combine to \( -7 \) when added together, resulting in a new term, \( -7 n^2 \). Remember, always maintain the variable part as it is when combining the coefficients. This simplification reduces clutter in your polynomial and prepares it for further operations.

Descending Order of Degree

Another key step in polynomial simplification is arranging the terms in descending order of degree. The degree of a term is determined by the exponent on its variable. The term with the highest degree should be placed first, followed by lower-degree terms in sequence. Using the same polynomial from our previous example, once we have combined like terms to get \( -7 n^2 -2 n + 7 \), we observe that \( -7 n^2 \) has a degree of 2 (since \( n \) is squared), \( -2 n \) has a degree of 1 (as \( n \) is to the power of 1), and \( 7 \) is a constant with a degree of 0.

By following this rule, we ensure the polynomial is neatly organized, making it easier to read and work with. The final organized polynomial, in this case, is: \( -7 n^2 -2 n + 7 \), which is exactly in descending order of degree from 2 to 0.

Polynomial Simplification

The overall process of polynomial simplification involves reducing a polynomial to its simplest form by combining like terms and ordering the terms properly. Simplification can make a polynomial easier to work with, whether for graphing, applying further algebraic operations, or solving equations. Taking the exercise we have \( -3 n^2 - 2 n + 7 - 4 n^2 \) and simplifying it, involves combining like terms to get \( -7 n^2 -2 n + 7 \), as discussed before.

Once the polynomial is simplified, it's crucial to check that all terms are in the correct order. As we’ve seen, this means descending order of degree. The simplest form for our exercise thus is \( -7 n^2 -2 n + 7 \). This clear, ordered, and reduced polynomial is what mathematicians and various mathematical applications need to proceed with further analysis or calculation.