Problem 64 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

After performing the indicated operations, you get the polynomial \(-5x^{4} - 4 xy - 9 y^{3}\) which is of the degree 4

Step 1: Rewrite the Subtraction

To simplify the process, rewrite the subtraction as addition by distributing the negative sign to each term in the second polynomial: \(x^{4}-7 x y-5 y^{3} + (-1)\times (6 x^{4}-3 x y+4 y^{3}) = x^{4}-7 x y-5 y^{3} - 6 x^{4}+3 x y-4 y^{3}\).

Step 2: Combine Like Terms

Combine the like terms. That is, \(x^{4} - 6x^{4}\), \(-7 xy + 3 xy\), and \(-5 y^{3} - 4 y^{3}\) yielding: \(-5x^{4} - 4 xy - 9 y^{3}\).

Step 3: Determine the Degree

The degree of a polynomial is the largest exponent in the polynomial. In this case, the term with the highest exponent is \(-5x^{4}\) giving us the polynomial degree of 4.

Key Concepts

Combining Like Terms

In algebra, combining like terms is a fundamental process to simplify expressions. Like terms are terms whose variables (and their exponents) are the same. For instance, in polynomial algebra, the terms \(3x^2\) and \(5x^2\) are like terms because they both contain the variable \(x\) raised to the same power of 2. On the other hand, \(3x^2\) and \(5x^3\) are not like terms because the exponents differ.

During polynomial subtraction, like terms are subtracted according to the usual rules of arithmetic. The coefficients of the terms are subtracted while the variable part remains unchanged. For a visual example, if we have \(x^4 - 6x^4\), we subtract the coefficients \(1 - 6\) to get \( -5\), and the simplified term is \( -5x^4\). It's important to ensure that only like terms are combined to avoid errors and correctly simplify the expression.

Polynomial Degree

The degree of a polynomial is a measure of its highest exponent when the polynomial is expressed in standard form (terms written from highest to lowest degree). The standard form is important because it shows the polynomial's degree clearly. For instance, the polynomial \(7x^3 - 2x^5 + 4\) would be written as \( -2x^5 + 7x^3 + 4\) in standard form, highlighting that the degree is 5.

Understanding polynomial degree is crucial because it can reveal key characteristics about the polynomial function, such as its possible number of roots, the general shape of its graph, and its end behavior as \(x\) approaches infinity or negative infinity. In our exercise, we determine that the resulting polynomial, after combining like terms, has the highest exponent of 4 in the term \( -5x^4\), which makes its degree 4.

Algebraic Operations

Algebraic operations refer to the different ways we can manipulate algebraic expressions, which include addition, subtraction, multiplication, division, and exponentiation. Each operation follows specific rules to ensure that the expressions are correctly simplified or solved.

In polynomial subtraction, as seen in our exercise, the operation involves a few steps, starting by expressing subtraction as the addition of a negative and then combining like terms. The ability to perform algebraic operations accurately is vital for success in mathematics, as these are the building blocks for solving equations, simplifying expressions, and understanding the relationships between different algebraic concepts.