Problem 56 - Elementary and Intermediate Algebra

Question

Subtract the polynomials. $$ \left(m n+8 n^{2}\right)-\left(6-5 m n+n^{2}\right) $$

Step-by-Step Solution

Answer

The result of subtraction is \(6mn + 7n^2 - 6\).

Step 1: Distribute the Negative Sign

Subtracting a polynomial is the same as distributing a negative sign and then adding. Begin by distributing the negative sign through the second polynomial: \[ (mn + 8n^2) - 6 + 5mn - n^2 \]

Step 2: Rearrange Terms

Rearrange the terms to group like terms together: \[ mn + 8n^2 + 5mn - n^2 - 6 \]

Step 3: Combine Like Terms

Combine like terms:For \(mn\): \[ mn + 5mn = 6mn \]For \(n^2\): \[ 8n^2 - n^2 = 7n^2 \]We then have: \[ 6mn + 7n^2 - 6 \]

Step 4: Write Final Answer

The final simplified expression is:\[ 6mn + 7n^2 - 6 \]

Key Concepts

Distribute Negative Sign

When subtracting polynomials, an important initial step is to correctly distribute the negative sign across the terms of the polynomial being subtracted. What does this mean? Essentially, you change the signs of each term in the polynomial you are subtracting.

For example, the original expression is \[(mn + 8n^2) - (6 - 5mn + n^2)\]In this case, we need to distribute the negative sign through the second polynomial, \[6 - 5mn + n^2\]This changes the expression to:

• \( -6 \) instead of \( 6 \)
• \( +5mn \) instead of \( -5mn \)
• \( -n^2 \) instead of \( +n^2 \)

Resulting in:\[mn + 8n^2 - 6 + 5mn - n^2\]Remember, distributing the negative sign is a pivotal part of polynomial subtraction, as it sets up the rest of the simplification process correctly.

Combine Like Terms

Once the negative sign has been properly distributed, the next step in polynomial subtraction is to combine like terms. But, what are like terms? Like terms are terms that have the same variable raised to the same power. So, essentially, you can only combine them if both these conditions meet.

In our example:\[mn + 8n^2 - 6 + 5mn - n^2\]We need to look for terms that are alike and add or subtract their coefficients:

• For \(mn\), combine \(mn\) and \(5mn\):\( mn + 5mn = 6mn \)
• For \(n^2\), combine \(8n^2\) and \(-n^2\):\( 8n^2 - n^2 = 7n^2 \)

Thus, after combining like terms, we achieve a simplified polynomial:\[6mn + 7n^2 - 6\]Combining like terms simplifies the expression, allowing you to see the reduced and more manageable form of your problem.

Simplified Expression

Finally, we arrive at obtaining the simplified expression after carrying out subtraction and simplification processes. The goal of simplifying is to present the polynomial in its most reduced and straightforward form.

In our example, after distributing the negative sign and combining like terms, we concluded with:\[6mn + 7n^2 - 6\]This expression is now streamlined and neatly combined with all its like terms. The simplified form is much easier to interpret and use in further calculations if needed.

Here are some key features of a simplified expression:

• All like terms are combined – ensuring no redundancy.
• Presented in an ordered form, often in descending powers of variables.
• Ready for any subsequent operations or evaluations you might want to undertake.

Remember, simplifying polynomials is crucial as it not only enables easier computation but also sharpens your algebraic skills.