Problem 66 - College Algebra

Question

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

Step-by-Step Solution

Answer

The resulting polynomial after performing the operation is \( 2x^{4}y^{2}+11x^{3}y-y-8x \) and the degree of this polynomial is 6.

Step 1: Arrange the Polynomials for Subtraction

First, organize the polynomials in the correct way for subtraction. Do this by placing like terms (terms that have the same variables and powers) above each other. The equation becomes: \( (5x^{4}y^{2}+6x^{3}y-7y) - (3x^{4}y^{2}-5x^{3}y-6y+8x) \).

Step 2: Perform the Subtraction

Subtract the terms in the second polynomial from the terms in the first polynomial. This results in a new polynomial: \( (5x^{4}y^{2}-3x^{4}y^{2}) + (6x^{3}y-(-5x^{3}y)) + (-7y-(-6y)) - 8x \). This simplifies to \( 2x^{4}y^{2}+11x^{3}y-y-8x \).

Step 3: Determine the Degree of the Polynomial

The degree of a polynomial is determined by the term with the highest power when considering the variables. In this polynomial, the term with the highest power is \( 2x^{4}y^{2} \). To find the degree, add the powers of \( x \) and \( y \) in this term, which is \( 4 + 2 = 6 \). So, the degree of the polynomial is 6.

Key Concepts

Polynomial Subtraction

Understanding polynomial subtraction involves removing or subtracting one polynomial from another. To begin, organize the given polynomials by aligning like terms—these are terms with identical variables raised to the same power. For instance, in the polynomial subtraction

• \((5x^{4}y^{2} + 6x^{3}y - 7y) - (3x^{4}y^{2} - 5x^{3}y - 6y + 8x)\),
• the like terms are \(5x^{4}y^{2}\) and \(3x^{4}y^{2}\), \(6x^{3}y\) and \(-5x^{3}y\), \(-7y\) and \(-6y\), and the unique term \(8x\).

Start by pairing these terms for subtraction, taking care to change the signs of the terms in the second polynomial before combining them.

For example:

• \(5x^{4}y^{2} - 3x^{4}y^{2} = 2x^{4}y^{2}\)
• \(6x^{3}y - (-5x^{3}y) = 11x^{3}y\)
• \(-7y - (-6y) = -y\)
• \(-8x\) remains as it is since there's no like term in the first polynomial.

This process of subtraction results in the simplified polynomial \(2x^{4}y^{2} + 11x^{3}y - y - 8x\). This method ensures a correct and streamlined way to work through polynomial subtraction problems.

Degree of a Polynomial

The degree of a polynomial is a crucial concept that tells us about the highest power of the polynomial's terms. Each term in a polynomial can have several variables, and the degree is determined by the term with the highest combined power.

In our example:

• The resulting polynomial is \(2x^{4}y^{2} + 11x^{3}y - y - 8x\).
• The degree is found in the term \(2x^{4}y^{2}\).

To calculate this, add the exponents of all the variables within the term. Here, the exponents are 4 for \(x\) and 2 for \(y\). Therefore, the degree is calculated as follows:

• \(4 + 2 = 6\)

This degree (6) tells us several things:

• The polynomial is of sixth degree.
• It can have at most 6 x-intercept crossings in its graph (though fewer is possible).
  

This awareness helps in predicting the behavior and potential complexity of polynomial equations.

Identifying Like Terms

Identifying like terms is a fundamental step in working with polynomials, whether you're adding, subtracting, or simplifying them.

Like terms are those that contain the same variables raised to the same powers. The coefficients can vary, but the variable part must be identical for the terms to be combined.

• For example, in our scenario the like terms include terms such as \(5x^{4}y^{2}\) and \(3x^{4}y^{2}\); they have similar variable factors, \(x^{4}y^{2}\).
• Similarly, the terms \(6x^{3}y\) and \(-5x^{3}y\) are like terms because they both involve \(x^{3}y\).

When performing operations such as addition or subtraction with polynomials, combining like terms helps simplify the expression. It reduces complexity and makes it clearer

• By seeing these terms as distinct groups, you can efficiently proceed with the required operations like subtraction.

Like terms are crucial in bringing order and simplicity to polynomial expressions, making them easier to handle and analyze.