Problem 64 - College Algebra

Question

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

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: Prioritize like terms

Identify like terms in both polynomials: \(x^4\) terms, \(xy\) terms and \(y^3\) terms.

Step 2: Perform Subtraction

Subtract the coefficients of the like terms. Hence, -7xy - (-3xy) = -7xy + 3xy = -4xy becomes the new variable xy term. Similarly, -5y^3 - 4y^3 = -9y^3 becomes the new y^3 term. Lastly, x^4 - 6x^4 = -5x^4 becomes the new x^4 term. Therefore, the combined polynomial after subtraction becomes \(-5x^4 -4xy - 9y^3\)

Step 3: Determining the Degree

The degree of the resulting polynomial is the highest exponent in the polynomial. Hence the degree of this polynomial is 4 which belongs to the term -5x^4.

Key Concepts

Degree of Polynomial

Understanding the degree of a polynomial is an essential skill in algebra. The degree of a polynomial is the highest power of the variable present in the expression. It essentially represents the term with the most significant "impact" in terms of growth rate as the variable increases.
For instance, in the polynomial expression resulting from a subtraction operation, \[-5x^4 - 4xy - 9y^3\]The degree is determined by identifying the term with the highest exponent of the variable 'x' or combination of variables. In this case, \(-5x^4\) has the highest power, which is 4. Hence, we say that this polynomial has a degree of 4.
Understanding the degree of a polynomial helps in comprehending the nature of the polynomial function, including its graph and end behavior.

Polynomial Subtraction

Polynomial subtraction involves subtracting two polynomial expressions, which is an operation similar to subtraction with regular numbers.
The key here is to focus on the coefficients of each term. Let’s revisit the polynomials from the exercise:\[(x^{4} - 7xy - 5y^{3}) - (6x^{4} - 3xy + 4y^{3})\]To subtract these, identify the coefficients of each like term (terms with the same variables raised to the same power) and then subtract accordingly:

• Subtract the coefficients of the \(x^4\) terms: \(1 - 6 = -5\)
• Subtract the coefficients of the \(xy\) terms: \(-7 - (-3) = -7 + 3 = -4\)
• Subtract the coefficients of the \(y^3\) terms: \(-5 - 4 = -9\)

After performing these operations, the resulting expression is: \[-5x^4 - 4xy - 9y^3\]Subtraction is all about ensuring each like term from one polynomial is directly aligned with its counterpart in the other polynomial.

Like Terms Identification

Identifying like terms is crucial when working with polynomials, as it is the first step in any polynomial operation such as addition, subtraction, or simplification.
Like terms refer to terms that contain the same variables raised to the same power. This means both the variables and the exponents of those variables must be identical for terms to be considered "like terms."
In our exercise, before performing any operations, the polynomials\[(x^{4} - 7xy - 5y^{3})\]and \[(6x^{4} - 3xy + 4y^{3})\]are analyzed to find like terms:

• Both have terms involving \(x^4\)
• Both have terms involving \(xy\)
• Both have terms involving \(y^3\)

By correctly identifying these like terms, one can proceed to either add or subtract their coefficients efficiently. Clear identification of like terms helps simplify the calculations and avoid errors.