Problem 37 - Algebra 1: Concepts and Skills

Question

Use a vertical format to add or subtract. $$ \left(3 x^{2}+7 x-6\right)-\left(3 x^{2}+7 x\right) $$

Step-by-Step Solution

Answer

The short answer to the problem is \(14x - 6\). This means, when \(3x^2 + 7x -6\) is subtracted by \(3x^2 +7x\), the result is \(14x - 6\).

Step 1: Write the expressions vertically

To start, express the problem in vertical format by writing each expression one under the other aligning the like terms. This would look as follows: \[ \begin{array}{c} 3 x^{2}+7 x-6 \ -3 x^{2}-7 x \ \end{array} \]

Step 2: Subtract coefficients of the same terms

After setting it up vertically, you subtract each term from the top expression to the similar term in the bottom expression. Remember, when subtracting, we add the opposite. Therefore, \(3x^2 - 3x^2 = 0\), and \(7x - (-7x) = 7x + 7x = 14x\). Now the equation would look as follows: \[ \begin{array}{c} 3 x^{2}+7 x-6 \ -3 x^{2}-7 x \ \hline 0 + 14x -6 \end{array} \]

Step 3: Simplify the result

After subtracting the coefficients of the same terms, simplify the result by removing any zero terms and combining like terms, yielding: \[ \begin{array}{c} 3 x^{2}+7 x-6 \ -3 x^{2}-7 x \ \hline 14x - 6 \end{array} \]

Key Concepts

Subtracting Polynomials Vertically

To understand subtracting polynomials vertically, imagine each term of the polynomial as a distinct item in a column. Similar to how you might stack numbers in a basic subtraction problem, polynomials can also be aligned vertically to make the subtraction process more organized and clear.

When setting up the problem, ensure that like terms are directly above or below each other. Like terms are terms that contain the same variables raised to the same power. For example, if you're trying to subtract \(3x^2\) from another \(3x^2\), these should be in the same vertical line. Once aligned, you simply subtract the coefficients (the numerical factors) of these like terms.

The process not only helps you keep track of your math but also visually emphasizes which terms cancel each other out and which ones need to be combined. This kind of visual organization is particularly helpful when the polynomials get long or complicated.

Combining Like Terms

When combining like terms, look for terms within an algebraic expression that have identical variable components. The idea is that only the coefficients of these similar terms can be added or subtracted. For instance, if you encounter \(14x - 6\), there's only one term with \(x\) and one constant term, so they are already simplified because they're not like terms.

However, if an equation had \(14x + 3x\), then these are like terms and can be combined to form \(17x\). Always remember, the variable part has to match exactly: \(x^2\) and \(x\) are not like terms because the exponents are different. Combining like terms is essential for simplifying algebraic expressions, making them easier to work with in subsequent calculations.

• Identify like terms.
• Add or subtract their coefficients.
• Keep the variable part unchanged.

Simplifying Algebraic Expressions

The final step in these processes often involves simplifying algebraic expressions. Simplification might sound just like cleaning up, but it's a crucial stage that involves ensuring that the expression is as straightforward as possible. This includes eliminating any zero terms that do not affect the value, like subtracting \(3x^2\) from \(3x^2\) to get 0, and then omitting the zero.

Any remaining like terms should be combined. This step is vital for making the expressions clear and ready for use in more complex calculations or equations. It's like tidying up your work so the most important parts stand out without any clutter. Be diligent in this step, as a well-simplified expression is both aesthetically pleasing and functionally efficient in mathematics. After simplifying, you may end up with an expression like \(14x - 6\), which has no like terms to combine and is therefore the simplest form of that particular expression.

Remember, a clear and simplified expression is the goal, making your work in mathematics accurate and impactful.