Problem 45 - Algebra for College Students

Question

Perform the operations as described. Subtract the sum of \(5 n^{2}-3 n-2\) and \(-7 n^{2}+n+2\) from \(-12 n^{2}-n+9\).

Step-by-Step Solution

Answer

\(-10n^2 + n + 9\)

Step 1: Find the sum of the given polynomials

We are given two polynomials: \(5n^2 - 3n - 2\) and \(-7n^2 + n + 2\). We need to add these two polynomials together. Combine the like terms: * For \(n^2\) terms: \(5n^2 + (-7n^2) = -2n^2\). * For \(n\) terms: \(-3n + n = -2n\). * For the constant terms: \(-2 + 2 = 0\).Thus, the sum of the polynomials is: \(-2n^2 - 2n\).

Step 2: Subtract the sum from the given polynomial

Now we need to subtract the sum \(-2n^2 - 2n\) from the polynomial \(-12n^2 - n + 9\). This means we will compute:\((-12n^2 - n + 9) - (-2n^2 - 2n)\)First, distribute the negative sign to the polynomial being subtracted:\(-12n^2 - n + 9 + 2n^2 + 2n\)Next, combine like terms:* For \(n^2\) terms: \(-12n^2 + 2n^2 = -10n^2\).* For \(n\) terms: \(-n + 2n = n\).* For constant terms: \(9\) remains unchanged as there is no other constant value.Thus, the expression simplifies to: \(-10n^2 + n + 9\).

Key Concepts

Subtracting Polynomials

When subtracting polynomials, the key is to carefully distribute the negative sign across the terms of the polynomial being subtracted. This involves changing the sign of each term in the polynomial that follows the subtraction symbol. Subtraction is essentially adding the opposite.
Let's apply this to the expression \((-12n^2 - n + 9) - (-2n^2 - 2n)\). Here is how to do this step by step:

• First, remove the parentheses and change the signs of the terms in \(-2n^2 - 2n\), resulting in \(+2n^2 + 2n\).
• Rewrite the expression as \(-12n^2 - n + 9 + 2n^2 + 2n\).

The next step is to combine like terms, which we will explain in another section, to arrive at the final expression of \(-10n^2 + n + 9\). Remember: distributing the negative sign is crucial to accurate polynomial subtraction.

Adding Polynomials

Adding polynomials is simpler than you might think. It's about pairing up terms that have the same variable raised to the same power, then adding their coefficients.
This allows you to combine them efficiently, making the polynomial simpler.
The concept is demonstrated well in our step where we first add \(5n^2 - 3n - 2\) and \(-7n^2 + n + 2\). Follow these steps:

• Identify like terms, such as those involving \(n^2\), those with \(n\), and constant terms.
• Add the coefficients of each identified group. Here, \(5n^2 + (-7n^2) = -2n^2\).
• Resulting conceptually in: \(-2n^2\) for squared terms.
• Repeat for linear and constant terms: \(-3n + n = -2n\) and \(-2 + 2 = 0\).

The sum simplifies the polynomials to \(-2n^2 - 2n\), showing how addition merges polynomial parts into a simpler form.

Combining Like Terms

Combining like terms involves grouping and merging terms that have the same variable parts. It's an essential step in simplifying polynomial expressions during addition or subtraction.
Here's how you do it effectively:

• First, identify terms that have the same variables and exponents. For example, \(n^2\) terms get grouped together, as do \(n\) terms and constants.
• Add or subtract their coefficients. Test pairs from our example show \(-12n^2\) and \(+ 2n^2\) combine to \(-10n^2\).
• Do the same for other terms: \(-n + 2n = n\) results in a single linear term, while standalone constants, like \(9\), are left as is.

The result, after combining, simplifies the expression to \(-10n^2 + n + 9\). This reduction of terms allows you to have a clearer view of the polynomial's components, aiding in further mathematical operations.