Problem 14 - Algebra for College Students

Question

Add the given polynomials. \(-7 t+14\) and \(-3 t-6\)

Step-by-Step Solution

Answer

The sum is \(-10t + 8\).

Step 1: Identify the Like Terms

In the given polynomials, like terms are the terms with the same variable raised to the same power. For the polynomials \(-7t + 14\) and \(-3t - 6\), the like terms are \(-7t\) and \(-3t\), and the constants \(14\) and \(-6\).

Step 2: Add the Like Terms

Add the like terms together separately. For the terms with \(t\), add \(-7t\) and \(-3t\) to get: \[-7t + (-3t) = -10t\]For the constant terms, add \(14\) and \(-6\) to get:\[14 + (-6) = 8\]

Step 3: Write the Resulting Polynomial

Combine the results from Step 2 to form the final polynomial. The sum of \(-10t\) and \(8\) gives:\[-10t + 8\]

Key Concepts

Like Terms

In polynomial mathematics, "like terms" are terms that have the same variable raised to the same power. Recognizing like terms is crucial when adding or subtracting polynomials. For example, in the polynomials \(-7t + 14\) and \(-3t - 6\), the expressions \(-7t\) and \(-3t\) are like terms because they both contain the variable \(t\) raised to the first power. The constant terms, like \(14\) and \(-6\), are also considered like terms even though they don't involve variables. Identifying these terms correctly will make the process of addition straightforward, as you will only combine terms that are similar, ensuring the polynomial structure is maintained.

Combining Like Terms

Combining like terms is a key step in simplifying polynomials. Once you've identified the like terms in your polynomials, you can add or subtract them. This helps in reducing the expression to its simplest form. Consider the terms \(-7t\) and \(-3t\): these can be combined by simply adding their coefficients together.

• Add the coefficients: \(-7 + (-3) = -10\)
• Retain the variable: \(-10t\)

This process shows how to effectively reduce the variable parts of a polynomial. Similarly, combine constant terms by direct addition:

• Constant terms: \(14 + (-6) = 8\)

This illustrates how combining like terms can simplify the process, setting the stage for straightforward polynomial addition.

Constant Terms

Constant terms are standalone numbers in a polynomial that do not change and have no variable attached to them. When performing operations on polynomials, such as addition, constant terms are simply added or subtracted directly. Take our example, \(14\) and \(-6\), where they are constant and do not involve any variables.These terms are often the easiest to combine because they require only basic addition or subtraction:

• Adding constants: \(14 + (-6) = 8\)

Understanding the role of constant terms is essential in polynomial operations because they directly impact the overall sum or difference in result. Their simplicity in manipulation helps make polynomial algebra more approachable, as they offer a clear cut operation compared to variable terms which necessitate attention to like terms.