- Group the given expression by its degree: \((x^2 - 6x^2) + (7x) + (-9 - 2)\).
- Combine the \(x^2\) terms: \(x^2 - 6x^2 = -5x^2\).
- Combine the \(x\) terms: \(7x + 0 = 7x\).
- Combine the constant terms: \(-9 - 2 = -11\).
- The simplified polynomial is \(\boxed{-5x^2 + 7x - 11}\).
Explanation
- Analysis of the Expression
The problem asks us to simplify the sum of two polynomials: \((x^2 + 7x - 9)\) and \((-6x^2 - 2)\) by combining like terms. Like terms are terms that have the same variables raised to the same power. To simplify the expression, we need to group these like terms together and perform the addition. In this expression, we have three types of terms: terms involving \(x^2\), terms involving \(x\), and constant terms.
- Combining Like Terms
First, we identify the terms involving \(x^2\). From the first polynomial, we have \(1x^2\), and from the second, we have \(-6x^2\). Adding these gives:
\[1x^2 + (-6x^2) = (1 - 6)x^2 = -5x^2.\]
Next, we identify the terms involving \(x\). The first polynomial contains \(7x\), and the second polynomial contains no \(x\)-term (which can be represented as \(0x\)). Adding these gives:
\[7x + 0x = 7x.\]
Finally, we identify the constant terms. The first polynomial has \(-9\), and the second has \(-2\). Adding these gives:
\[-9 + (-2) = -11.\]
- Final Simplified Polynomial
Now, we combine the results from the previous step to form the simplified polynomial:
\[-5x^2 + 7x - 11.\]
This is the simplified form of the sum \((x^2 + 7x - 9) + (-6x^2 - 2)\).
Examples
Combining like terms is a fundamental skill in algebra used whenever you need to simplify expressions in physics, economics, or engineering. For example, if you are modeling the total cost of producing a product, you might have fixed costs and variable costs expressed as a polynomial. Simplifying this polynomial helps you quickly understand how the costs behave as the production quantity (\(x\)) changes, allowing for easier budget planning and analysis.