Problem 66 - College Algebra Essentials

Question

Perform the indicated operations. Indicate the degree of the resulting polynomial. $$\left(5 x^{4} y^{2}+6 x^{3} y-7 y\right)-\left(3 x^{4} y^{2}-5 x^{3} y-6 y+8 x\right)$$

Step-by-Step Solution

Answer

The resulting polynomial after performing the indicated operations is \(2 x^{4} y^{2} + 11 x^{3} y - y - 8 x\), and its degree is 6.

Step 1: Identify Like Terms

In each polynomial, terms that have the same variables and powers are called 'like terms'. Therefore, first we will identify 'like terms' in both polynomials. 'Like terms' in our expression are \(5 x^{4} y^{2}\) and \(3 x^{4} y^{2}\), \(6 x^{3} y\) and \(-5 x^{3} y\), \(-7 y\) and \(-6 y\), and finally \(8 x\), although this term doesn't have a counterpart in the first polynomial.

Step 2: Perform the Subtraction

Subtract the corresponding coefficients of the 'like terms'. First, \(5 x^{4} y^{2} - 3 x^{4} y^{2} = 2 x^{4} y^{2}\). Second, \(6 x^{3} y-(-5 x^{3} y) = 6 x^{3} y + 5 x^{3} y = 11 x^{3} y\). Third, \(-7 y -(-6 y) = -7 y + 6 y = -y\). Finally, since \(8 x\) does not have a counterpart in the first polynomial, it simply turns into \(-8 x\) when subtracted.

Step 3: Write Down the Resulting Polynomial and Identify Its Degree

Write down the resulting polynomial, which is \(2 x^{4} y^{2} + 11 x^{3} y-y -8x \). The degree of a polynomial is the highest sum of the powers of its variables in any term. Looking at each term in our polynomial, we see that the first term has degree 4+2=6. All other terms have a degree less than 6. Therefore, the degree of the resulting polynomial is 6.

Key Concepts

Like Terms

In polynomial operations, understanding the concept of 'like terms' is crucial. Like terms are terms within a polynomial that have the same variables raised to the same power. For example, in the polynomials provided in our exercise, terms such as \(5x^4y^2\) and \(3x^4y^2\) are like terms because they both have the variables \(x^4\) and \(y^2\). Similarly, \(6x^3y\) and \(-5x^3y\) are like terms as both have the same variable with the same power structure, \(x^3y\).

Recognizing like terms allows us to accurately combine them by performing addition or subtraction operations on their coefficients. For instance, when dealing with like terms such as \(6x^3y\) and \(-5x^3y\), we combine their coefficients (6 and -5), leaving the variables unchanged. This results in a new term \(x^3y\), with a combined coefficient of 1, although in our calculation the coefficient becomes 11 due to prior operations.

By carefully identifying and combining like terms, the simplification of polynomials becomes a more manageable task, allowing for easier subsequent operations.

Polynomial Subtraction

Polynomial subtraction involves the systematic removal or deduction of one polynomial from another. When subtracting polynomials, it is essential to ensure that subtraction is performed only on like terms. In our exercise, we subtract corresponding coefficients of like terms in the expressions.

• Subtract \(3x^4y^2\) from \(5x^4y^2\) to get \(2x^4y^2\)
• Subtract \(-5x^3y\) from \(6x^3y\), effectively adding \(5x^3y\) to get \(11x^3y\)
• Subtract \(-6y\) from \(-7y\), leading to subtracting \(-6y\) which is the same as adding \(6y\), resulting in \(-y\)
• The term \(8x\) has no like term in the first polynomial, so subtracting \(8x\) results in \(-8x\)

When performing polynomial subtraction, taking care to align like terms vertically can help avoid errors. This systematic elimination preserves the structure of the final polynomial, while proper attention to signs ensures accurate computation.

Degree of a Polynomial

The degree of a polynomial is a key characteristic, determining the maximum value of the sum of the exponents in a term. It gives insight into the polynomial's behavior and is essential when speaking about polynomial size and complexity.

In our resulting polynomial \(2x^4y^2 + 11x^3y - y - 8x\), the degree is determined by looking at each term separately. We calculate:

• For \(2x^4y^2\), the degree is \(4+2=6\), making it the highest degree term in the polynomial.
• \(11x^3y\) has a degree of \(3+1=4\).
• The degree of \(-y\) is 1.
• \(-8x\) has a degree of 1.

The overall degree of a polynomial is interpreted as the highest degree among its terms. Here, our polynomial's degree is 6, deriving from the term \(x^4y^2\), because it contributes the largest sum of variable powers. Understanding the degree is crucial in applications involving polynomial fits, roots, and even asymptotic behavior.