- Distribute the negative sign to the second polynomial: \((-2x^2 + 10x + 6) + 8x^2 + 10\).
- Group like terms by power: \((-2x^2 + 8x^2) + 10x + (6 + 10)\).
- Add the coefficients for each term: \(6x^2 + 10x + 16\).
- The final simplified result is \(\boxed{6x^2 + 10x + 16}\).
Explanation
- Analysis of the Polynomial Subtraction
We are asked to perform the subtraction of two polynomials: \((-2x^2 + 10x + 6)\) and \((-8x^2 - 10)\). Subtracting one polynomial from another is equivalent to adding the opposite of the second polynomial to the first. We can visualize this process using the concept of algebra tiles, where we have tiles representing \(x^2\), \(x\), and constants. Negating a polynomial means changing the sign of each of its terms.
- Distributing the Negative Sign
To subtract the polynomials, we start by distributing the negative sign across every term in the second polynomial. The expression \((-2x^2 + 10x + 6) - (-8x^2 - 10)\) becomes \(-2x^2 + 10x + 6 + 8x^2 + 10\). Distributing the negative sign essentially "flips" the signs of the terms \(-8x^2\) and \(-10\), turning them into \(+8x^2\) and \(+10\), respectively.
- Grouping Like Terms
Now that all terms are expressed as an addition, we group the terms that share the same power of \(x\) (the like terms). We have:
- The \(x^2\) terms: \(-2x^2 + 8x^2\)
- The \(x\) terms: \(10x\)
- The constant terms: \(6 + 10\)
Grouping these gives:
\((-2x^2 + 8x^2) + 10x + (6 + 10)\).
- Final Simplified Polynomial
Finally, we combine the coefficients of the like terms:
- For \(x^2\): \(-2 + 8 = 6\), resulting in \(6x^2\).
- For \(x\): there is only \(10x\).
- For the constant: \(6 + 10 = 16\).
Putting it all together, the simplified polynomial is \(6x^2 + 10x + 16\).
Examples
Algebra tiles are a great visual tool for understanding how expressions work! For example, if you are calculating the profit of a business where the initial setup cost is \(-8x^2 - 10\) and your revenue is \(-2x^2 + 10x + 6\), subtracting the cost from the revenue helps determine your net gain. By grouping the terms just like we did with the tiles, you can easily see how different cost and revenue components cancel out or combine to give your final financial result.