Problem 27 - College Algebra with Modeling and Visualization

Question

Add the polynomials. $$\left(2 z^{3}+5 z-6\right)+\left(z^{2}-3 z+2\right)$$

Step-by-Step Solution

Answer

The sum of the polynomials is \(2z^3 + z^2 + 2z - 4\).

Step 1: Identify Like Terms

List out each polynomial separately. From the first polynomial \(2z^3 + 5z - 6\), identify the terms: \(2z^3\), \(+5z\), and \(-6\). From the second polynomial \(z^2 - 3z + 2\), identify these terms: \(z^2\), \(-3z\), and \(+2\). Like terms are found amongst corresponding powers of \(z\).

Step 2: Rewrite and Group Like Terms

Write the combined polynomials: \(2z^3 + 5z - 6 + z^2 - 3z + 2\). Group the like terms together: \(2z^3\) with no similar term, \(+z^2\) with no similar term, \(+5z\) and \(-3z\), \(-6\) and \(+2\). It appears as: \(2z^3 + z^2 + (5z - 3z) + (-6 + 2)\).

Step 3: Perform Addition on Like Terms

Add the like terms: \(2z^3\) remains the same, \(+z^2\) remains the same, combine \(+5z\) and \(-3z\) to get \(+2z\), and combine \(-6\) and \(+2\) to get \(-4\). So, the resulting terms are: \(2z^3 + z^2 + 2z - 4\).

Step 4: Write the Final Polynomial

Combine all the terms from the addition: \(2z^3 + z^2 + 2z - 4\). Ensure all terms are simplified and presented correctly, confirming all like terms are combined.

Key Concepts

Like Terms

In the world of polynomials, recognizing **like terms** is crucial. Like terms are those polynomial terms that have the same variables raised to the same powers. This means, for example, that in the expressions involving the variable \(z\), terms like \(5z\) and \(-3z\) are like terms because they both involve \(z\) raised to the first power. Here’s how to spot and think about like terms:

- **Same variable and exponent:** Only terms with identical variable parts can be combined. \(5z\) and \(-3z\) both have \(z^1\).
- **Doesn’t matter what number:** The coefficients (the numbers in front of the variables) can be different. You simply add or subtract these once you've grouped the like terms.
- **Constant terms are like terms:** Numbers without variables, like \(-6\) and \(+2\), are also considered like terms because they're constants.

Spotting like terms is often the first step in polynomial operations like addition and subtraction, setting the stage for simplifying expressions.

Combining Like Terms

Once you’ve identified the like terms, the next step is **combining** them. This process helps reduce the polynomial to its simplest form by adding or subtracting coefficients of the like terms.

- **Grouping for clarity:** Write each polynomial term next to its like terms. For example, \(5z\) and \(-3z\) can be combined because they are both terms of \(z\) to the first power. The combined result is \(5z + (-3z) = 2z\).
- **Dealing with constants:** Similarly, handle numbers without variables as like terms. Combine \(-6\) and \(+2\) to get \(-4\).
- **No like terms? No problem!** If there are no other like terms to combine, for instance \(2z^3\) or \(z^2\), these terms remain as they are.

This step simplifies your polynomial, making it easier to understand and use in further calculations.

Polynomial Simplification

**Simplifying a polynomial** involves turning a complex expression into one that is more straightforward and easier to work with. It often involves using the processes of identifying and combining like terms. Here’s what to focus on during simplification:

- **Order and cleanliness:** Always write the polynomial terms in descending order of their exponents. For instance, \(2z^3 + z^2 + 2z - 4\) is ordered correctly from highest exponent to lowest.
- **Final check:** Once all operations are completed, double-check to ensure all possible like terms have been combined, and the expression cannot be simplified further.
- **Applications and further use:** A simplified polynomial is not only tidier but is also ready for further mathematical operations like differentiation or integration if necessary.

Simplification makes your polynomials more manageable and opens the door to more advanced algebraic manipulation.