Problem 65 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The resulting polynomial after performing the indicated operations is \(x^{4}y^{2} + 8x^{3}y + y - 6x\) and its degree is 6.

Step 1: Distribute the Negative Sign

Distribute the negative sign from the second set of parentheses to every term inside, which transforms the second polynomial into \(-2 x^{4} y^{2}+3 x^{3} y+4 y-6 x\). So the expression now reads: \(3 x^{4} y^{2}+5 x^{3} y-3 y -2 x^{4} y^{2}+3 x^{3} y+4 y-6 x\)

Step 2: Combine Like Terms

Now add and subtract the similar terms in the expression. This gives: \((3x^{4}y^{2} - 2x^{4}y^{2}) + (5x^{3}y + 3x^{3}y) + (-3y + 4y) - 6x \). When simplified, the expression becomes: \(x^{4}y^{2} + 8x^{3}y + y - 6x\).

Step 3: Find the Degree of the Polynomial

The degree of a polynomial is the sum of the exponents of all variables in a term that has the highest total degree. We have three terms: \(x^{4}y^{2}\) with degree 6, \(8x^{3}y\) with degree 4, and \(- 6x\) with degree 1. Therefore, the degree of the resulting polynomial is 6.

Key Concepts

Degree of a Polynomial

Understanding the degree of a polynomial is crucial as it indicates the highest power present within the expression. The degree is determined by identifying the term with the greatest sum of exponents.
For instance, in the expression \(3x^{4}y^{2}\), the degree is the sum of the exponents of both variables \(x\) and \(y\), which is \(4 + 2 = 6\). In any polynomial expression, you look for the highest degree term. This highest degree gives the overall degree of the polynomial.
Let's consider our simplified polynomial \(x^{4}y^{2} + 8x^{3}y + y - 6x\). Here, \(x^{4}y^{2}\) has the highest degree of 6, making it the degree of the entire polynomial. Recognizing this helps us focus on the most significant term in polynomial operations.

Combining Like Terms

Combining like terms simplifies polynomial expressions. It involves adding or subtracting terms with the same variables raised to the same power.
In the given exercise, start by identifying terms that share the same variable components. For example, both \(3x^{4}y^{2}\) and \(-2x^{4}y^{2}\) are like terms since they contain identical variables raised to the same exponents. By performing \(3x^{4}y^{2} - 2x^{4}y^{2}\), it results in just \(x^{4}y^{2}\).

• For \(5x^{3}y\) and \(3x^{3}y\), adding them gives \(8x^{3}y\).
• Terms \(-3y\) and \(4y\) simplify to \(y\).

This reduction not only simplifies the expression but also reveals clearer insights about its structure and behavior.

Distributive Property

The distributive property is a fundamental operation that lets us simplify expressions by breaking down complex expressions into manageable parts. When subtracting a polynomial, distributing the negative sign ensures that each term in the parentheses is effectively subtracted.
In the original problem, the second polynomial \(2x^{4}y^{2} - 3x^{3}y - 4y + 6x\) is subtracted from the first. By distributing the negative sign to each term, it transforms into \(-2x^{4}y^{2} + 3x^{3}y + 4y - 6x\). This step is crucial for facilitating accurate term combination.
Overall, applying the distributive property in polynomial operations uncovers all terms for straightforward processing, ensuring errors are minimized during simplification.