91Ó°ÊÓ

When subtracting polynomials, the first step is to distribute the negative sign to each term of the polynomial being subtracted. This helps ensure each term's sign is correctly adjusted before any further computation. Let's take a closer look at this concept with our example:

We have: \((6a^{2} + 5ab - 4b^{2}) - (8a^{2} - 7ab + 3b^{2})\)

To distribute the negative sign, multiply each term inside the second polynomial by -1:
\[ - (8a^{2} - 7ab + 3b^{2}) = -8a^{2} + 7ab - 3b^{2} \]
After distributing the negative sign, the expression becomes:
\[ 6a^{2} + 5ab - 4b^{2} - 8a^{2} + 7ab - 3b^{2} \]
This step is crucial as it changes the signs of all terms in the second polynomial, allowing us to proceed with combining like terms.

Once the negative sign is distributed, the next step is to combine like terms. Like terms are terms that have identical variables raised to the same power. Combining them involves adding or subtracting their coefficients:

After distributing the negative sign, we have:
\[ 6a^{2} + 5ab - 4b^{2} - 8a^{2} + 7ab - 3b^{2} \]
Now, let's combine the like terms:
- Combine the \(a^{2}\) terms:
\[ 6a^{2} - 8a^{2} = -2a^{2} \]
- Combine the \(ab\) terms:
\[ 5ab + 7ab = 12ab \]
- Combine the \(b^{2}\) terms:
\[ -4b^{2} - 3b^{2} = -7b^{2} \]
Combining like terms simplifies our polynomial and prepares it for the final expression.

The final step in polynomial subtraction is to simplify the expression by combining the results of the like terms:

We've already distributed the negative sign and combined the like terms. Now, we bring everything together:
From the combined terms, we have:
\[ -2a^{2} + 12ab - 7b^{2} \]
This is the simplified expression of the original polynomial subtraction problem. By carefully distributing the negative sign, combining like terms, and simplifying, you can solve any polynomial subtraction problem step-by-step.