Problem 30 - Elementary and Intermediate Algebra

Question

Add the polynomials. $$ \left(2 t^{2}+11 t-15\right)+\left(-5 t^{2}-13 t+10\right) $$

Step-by-Step Solution

Answer

The sum of the polynomials is \(-3t^2 - 2t - 5\).

Step 1: Write down the polynomials

First, write down each polynomial with all of its terms. Start with \(2t^2 + 11t - 15\) and then \(-5t^2 - 13t + 10\).

Step 2: Combine like terms

Now, combine the like terms from each polynomial. Like terms are terms that have the same variables raised to the same power.1. Combine the \(t^2\) terms: \(2t^2 + (-5t^2) = -3t^2\).2. Combine the \(t\) terms: \(11t + (-13t) = -2t\).3. Combine the constant terms: \(-15 + 10 = -5\).

Step 3: Write the final polynomial

After combining all like terms, write down the final polynomial as the sum of the combined terms. This results in:\(-3t^2 - 2t - 5\).

Key Concepts

Adding Polynomials

Adding polynomials is straightforward once you grasp the core idea that you're not just lumping numbers together, but matching 'like' items. Each term in a polynomial is made up of a coefficient and variable(s). When adding polynomials, you write down all the terms from each polynomial and group them in parentheses, just as shown in our initial exercise. It is crucial to keep an eye on signs (positive and negative) because mixing them up can lead to errors.
To add polynomials:

• Write each polynomial, ensuring all like terms are directly one above the other if doing by hand.
• Add coefficients of the same degree terms, keeping variables and exponents constant.
• Combine results for the final solution.

This might sound a bit complex, but with practice, it becomes second nature. Notably, adding polynomials can be done vertically like classic arithmetic addition, or horizontally as shown in the example.

Like Terms

When dealing with polynomials, understanding what like terms are is crucial. Like terms are terms whose variables (and their exponents) are the same. They do not need to have the same coefficients; only the variables and their powers must be identical for terms to be considered 'like'.
For example:

• In the expression \(3x^2 + 2x - 5\), the like terms are grouped by variables and exponents:
  • \(3x^2\) and any other \(x^2\) term would be a pair of like terms.
  • \(2x\) is similar to any other terms with just \(x\), like \(-3x\).
  • \(-5\) is a constant and is paired with other constants.

Finding like terms helps in efficiently simplifying expressions. If terms are not like, they cannot be combined, and must stay as separate parts of the polynomial. Remember, the power and variable must match for terms to be combined effectively.

Combining Like Terms

Combining like terms involves consolidating terms that have identical variables and exponents into a single term. Consider this powerful tool the key to simplifying polynomials, which then allows for easier manipulation in any further calculations or applications.
Here's how to combine like terms effectively:

• Identify all terms that have the same variable and exponent.
• Add or subtract the coefficients of these terms.
• Write down the result as one consolidated term.

In the problem at hand, we follow each step to simplify, based on their like terms within the polynomials being added:

• For \(t^2\) terms: Combine \(2t^2\) and \(-5t^2\), resulting in \(-3t^2\).
• For \(t\) terms: Assemble \(11t\) and \(-13t\), resulting in \(-2t\).
• For constant numbers: Add \(-15\) and \(10\), resulting in \(-5\).

This breakdown ensures the most simplified form of the polynomial, \[-3t^2 - 2t - 5\], offering a clean, concise outcome.