Problem 62 - College Algebra Essentials

Question

Perform the indicated operations. Indicate the degree of the resulting polynomial. $$\left(7 x^{4} y^{2}-5 x^{2} y^{2}+3 x y\right)+\left(-18 x^{4} y^{2}-6 x^{2} y^{2}-x y\right)$$

Step-by-Step Solution

Answer

The result of the operation is \(-11x^{4}y^{2} - 11x^{2}y^{2} + 2xy\) and the degree of the resulting polynomial is 6.

Step 1: Identify Like Terms

Like terms are terms that have the same variables raised to the same power. In this exercise the like terms are \(7x^{4}y^{2}\) and \(-18x^{4}y^{2}\), \(-5x^{2}y^{2}\) and \(-6x^{2}y^{2}\), and \(3xy\) and \(-xy\).

Step 2: Add Like Terms

To add like terms, simply add the coefficients of the like terms. The coefficients of \(7x^{4}y^{2}\) and \(-18x^{4}y^{2}\) are 7 and -18, which when added give \(-11x^{4}y^{2}\). The coefficients of \(-5x^{2}y^{2}\) and \(-6x^{2}y^{2}\) are -5 and -6, when added, result in \(-11x^{2}y^{2}\). Finally, the coefficients of \(3xy\) and \(-xy\) are 3 and -1 which when added result in \(2xy\).

Step 3: Identify the Degree of the Resulting Polynomial

After performing the operations, the resulting polynomial is \(-11x^{4}y^{2} - 11x^{2}y^{2} + 2xy\). The degree of the polynomial is the largest sum of exponents of variables in a term. In this case, the first term has degree 4+2=6, the second term has degree 2+2=4, and the last term has degree 1+1=2. Therefore, the degree of the polynomial is 6, which is the largest sum of exponents.

Key Concepts

Understanding Like Terms

In polynomials, like terms make things simpler to work with. Like terms have exactly the same variables, each raised to identical powers. For instance, in the expression \(7x^{4}y^{2}\) and \(-18x^{4}y^{2}\), both terms have the same variables \(x\) and \(y\), with \(x\) raised to the power of 4 and \(y\) to the power of 2. This means they are like terms. Working with like terms helps in simplifying polynomials.
By identifying like terms, you can focus on the coefficients and avoid getting tangled in the variables. Remember, the key is ensuring the variables and their powers must match exactly. For example:

• \(7x^{4}y^{2}\) and \(-18x^{4}y^{2}\) are like terms.
• \(-5x^{2}y^{2}\) and \(-6x^{2}y^{2}\) are like terms.
• \(3xy\) and \(-xy\) are like terms.

Recognizing like terms is the first step in performing polynomial operations.

Degree of a Polynomial Clarified

The degree of a polynomial is a way to determine its 'power'. It is the highest sum of the exponents of variables in any single term within the polynomial. This tells us essentially how 'large' or 'powerful' the component of the term with the greatest impact is.
For example, let's look at some terms from our original exercise:

• The term \(-11x^{4}y^{2}\) has a degree of \(4 + 2 = 6\) because it has two variables where \(x\) is raised to the 4th power and \(y\) is raised to the 2nd power.
• For \(-11x^{2}y^{2}\), the degree is \(2 + 2 = 4\).
• Lastly, \(2xy\) has a degree of \(1 + 1 = 2\).

Identifying the degree is crucial for understanding the hierarchy of terms and their influence in the polynomial as a whole.

Adding Polynomials Made Easy

When it comes to adding polynomials, you're simply merging like terms. This involves adding or subtracting their coefficients while keeping the same variable parts.
Let's break it down using our example:

• From \(7x^{4}y^{2}\) and \(-18x^{4}y^{2}\), we take the coefficients 7 and -18, which add up to give \(-11\). So their addition is \(-11x^{4}y^{2}\).
• Next, adding \(-5x^{2}y^{2}\) and \(-6x^{2}y^{2}\) results in \(-11x^{2}y^{2}\) because their coefficients are -5 and -6.
• Lastly, combine \(3xy\) and \(-xy\) by adding their coefficients 3 and -1 to get \(2xy\).

Always ensure that when adding polynomials, you're handling like terms. This means taking extra care to align terms with identical variables and power expressions. Once all like terms are combined, you have your simplified polynomial.