Problem 82 - Beginning and Intermediate Algebra

Question

Each of the polynomials below is a polynomial in two variables. Perform the indicated operation(s). $$\left(-8 u^{2} v^{2}+2 u v+3\right)-\left(-9 u^{2} v^{2}-14 u v+18\right)$$

Step-by-Step Solution

Answer

The short answer to the subtraction operation between the given polynomials is: \(u^{2}v^{2} + 16uv - 15\).

Step 1: Write down the given polynomials

We are given the following two polynomials and we need to perform the subtraction operation between them: $$\left(-8u^{2}v^{2} + 2uv + 3\right) - \left(-9u^{2}v^{2} - 14uv + 18\right)$$

Step 2: Distribute the negative sign across the second polynomial

We will distribute the negative sign across the terms in the second polynomial as follows: $$\left(-8u^{2}v^{2} + 2uv + 3\right) + \left(9u^{2}v^{2} + 14uv - 18\right)$$

Step 3: Combine like terms

Now, we combine the like terms (the terms having the same powers of u and v) in the expression: $$(-8u^{2}v^{2} + 9u^{2}v^{2}) + (2uv + 14uv) + (3 - 18)$$

Step 4: Add the like terms

Next, we add the like terms to simplify the expression: $$1u^{2}v^{2} + 16uv - 15$$

Step 5: Write down the final answer

The final answer after performing the subtraction operation between the two given polynomials is: $$u^{2}v^{2} + 16uv - 15$$

Key Concepts

Two Variables

In mathematics, polynomials can involve one or more variables. Here, we have a polynomial with two variables, \(u\) and \(v\). Each term in the polynomial contains a combination of these variables raised to different powers. This allows us to describe more complex relationships between two quantities. By using polynomials with two variables, we can model surfaces or regions in a two-dimensional plane, providing a rich visual representation in certain applications. When working with such polynomials, it is crucial to keep track of both variables and their respective powers.

Subtraction Operation

Subtraction in polynomials involves taking the second polynomial and subtracting it from the first. This operation is similar to regular subtraction but occurs term by term. To perform this operation correctly:

• Align the polynomials so that each term matches with its corresponding term in terms of the powers of the variables involved.
• Subtract coefficients of like terms, which are terms with the same combination of variables and powers.

It's important to ensure accuracy in aligning and subtracting to avoid errors, resulting in a new polynomial expression.

Combine Like Terms

Combining like terms is a fundamental step in simplifying polynomials. Like terms are those that have the same variable raised to the same power in each polynomial involved. For example, terms like \(-8u^2v^2\) and \(9u^2v^2\) are like terms because they both involve \(u^2v^2\).

• Identify all pairs or groups of like terms across the polynomials.
• Add their coefficients together to form a single term.

This process helps in reducing the complexity of the polynomial and in obtaining a simplified equation that is much easier to work with.

Distribute Negative Sign

Distributing the negative sign is a critical step when dealing with subtraction between polynomials. This step ensures that each term in the second polynomial changes its sign before being combined with the first polynomial. To distribute the negative sign:

• Multiply each term in the second polynomial by \(-1\).
• This changes the sign of each coefficient within the polynomial.

For instance, \(-(-9u^2v^2 - 14uv + 18)\) becomes \(9u^2v^2 + 14uv - 18\) after distributing the negative sign. This conversion prepares the expression for combining like terms effectively.