Problem 64 - Algebra and Trigonometry

Question

In Exercises 59–66, 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

The resulting polynomial after subtraction is \(-5x^{4} -4xy - 9y^{3}\) and the degree of the polynomial is 4.

Step 1: Group Similar Terms

The first step is to group similar terms from the polynomials together. Similar terms are the ones that have the same variable parts. So here, group like terms of \(x^{4}\), \(x y\), and \(y^{3}\) together. \[ (x^{4} - 6x^{4}) + (-7xy - -3xy) + (-5y^{3} - 4y^{3}) \]

Step 2: Perform The Subtraction

The second step entails performing the subtraction operation on grouped terms. Here, subtract the coefficients of like terms. \[ -5x^{4} -4xy - 9y^{3} \]

Step 3: Find The Degree of Resulting Polynomial

Now, find the degree of the resulting polynomial. The degree is the highest power of the variable in the polynomial. In this case, the highest power is 4 in the term \(-5x^{4}\). So, the degree of the polynomial is 4.

Key Concepts

Degree of a Polynomial

In understanding polynomials, the degree is a fundamental concept. It represents the highest power of the variable within a polynomial expression. For example, in the polynomial \(-5x^4 - 4xy - 9y^3\), the term with the highest degree is \(-5x^4\). Here, the power of \(x\) is 4, making 4 the degree of the polynomial.

Why is this important? The degree provides insights into the polynomial's behavior, such as the number of roots or the shape of its graph. Always look for the largest exponent to determine the degree.

Like Terms

Like terms are terms in a polynomial that have identical variable parts raised to the same power. Recognizing like terms is crucial when simplifying or performing operations on polynomials. For example, in the expression \(x^{4}-6x^{4}\), both terms involve \(x^{4}\) and can be combined.

• Terms like \(-7xy\) and \(-3xy\) are also like terms because they both involve \(xy\).
• Terms like \(-5y^3\) and \(4y^3\) together can be combined as they share \(y^3\).

By grouping and combining these terms, you can simplify complex expressions easily.

Polynomial Operations

Polynomial operations involve various functions like addition, subtraction, multiplication, and division. Mastering these operations is key to solving polynomial equations. In subtraction, like in our example, the task is to subtract coefficients of similar terms.

Steps to perform polynomial subtraction:

• Identify like terms between the polynomials.
• Substitute and combine the coefficients. For example, \(x^4 - 6x^4 = -5x^4\).

This operation allows for simplification, revealing the essential polynomial structure and degree.