Problem 65 - College Algebra

Question

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

Step-by-Step Solution

Answer

The simplified polynomial is \(x^{4}y^{2} + 8x^{3}y - y - 6x\) and the degree of the polynomial is 6.

Step 1: Understand the operation

Here, the operation is subtraction between two polynomials: \(3x^{4}y^{2} + 5x^{3}y - 3y\) and \(2x^{4}y^{2} - 3x^{3}y - 4y + 6x\). The key is to simplify the operation by combining like terms.

Step 2: Apply the operation

We subtract the second polynomial from the first: \((3x^{4}y^{2} + 5x^{3}y - 3y) - (2x^{4}y^{2} - 3x^{3}y - 4y + 6x)\). This leads us to \(3x^{4}y^{2} - 2x^{4}y^{2} + 5x^{3}y - (-3x^{3}y) - 3y - (- 4y) - 6x\).

Step 3: Simplify the resulting polynomial

Now, we simplify the polynomial by combining like terms, it results in: \(x^{4}y^{2} + 8x^{3}y - y - 6x\).

Step 4: Identify the degree of the polynomial

The degree of a polynomial is the highest degree of its terms, in this polynomial, the term with highest degree is \(x^{4}y^{2}\), so the degree of the polynomial is 6 (sum of the exponents of x and y).

Key Concepts

Degree of Polynomial

In the study of polynomials, understanding the degree is crucial. The degree of a polynomial is simply the highest sum of the exponents of variables in a term. Consider each term in a polynomial individually and add the powers of the variables. For example, in the term \(x^{4}y^{2}\), the degree is calculated by adding the exponents of \(x\) and \(y\), which equals 6 (4 + 2).

This indicates that the degree of the polynomial determines the polynomial's behavior and growth rate. The degree shows the highest "dimension" of the polynomial and tells us about the number of roots and the possible turning points on its graph.

Like Terms

Like terms are a key part of simplifying polynomials. They are terms that have exactly the same variable parts, with the same exponents. It doesn’t matter what the coefficients are; if the variable parts match, they're like terms.

For example, in the polynomial \(3x^{4}y^{2} + 5x^{3}y - 3y - 2x^{4}y^{2} + 3x^{3}y + 4y - 6x\), we have the like terms: \(3x^{4}y^{2}\) and \(-2x^{4}y^{2}\), \(5x^{3}y\) and \(3x^{3}y\), as well as \(-3y\) and \(4y\).

Combining like terms simplifies polynomials, as seen above, where these terms are added (or subtracted) by summing their coefficients.

Simplifying Expressions

Simplifying expressions in polynomial operations involves a few clear steps. Begin by performing operations such as addition or subtraction. Here, you must ensure each operation is computed correctly by distributing negatives or considering parentheses first.

This means subtracting the entire second polynomial from the first often results in changing signs inside the second polynomial. Once the correct operations are set up mathematically, identify and combine like terms.

• Perform each specified arithmetic operation step-by-step.
• Combine coefficients of like terms for simplification.

Finally, organize the terms in descending order of their degrees to clearly see the simplified polynomial expression. This makes it easy to identify the highest degree of the polynomial, which leads us back to the original concept of the degree.