Problem 66 - Algebra and Trigonometry

Question

In Exercises 59–66, 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 result of the operation is the polynomial \(2x^{4}y^{2} + 11x^{3}y - y - 8x\), and the degree of the resulting polynomial is 4.

Step 1: Change the signs

Change the sign of each term of the polynomial that is being subtracted. The equation then transforms to: \[ \left(5 x^{4} y^{2}+6 x^{3} y-7 y\right) + \left(-3 x^{4} y^{2}+5 x^{3} y+6 y-8 x\right) \]

Step 2: Combine like terms

Now, add these two polynomials by combining like terms together. The like terms are those which have the same variables raised to the same power. The result will be: \[ (5x^{4}y^{2} - 3x^{4}y^{2}) + (6x^{3}y + 5x^{3}y) + (-7y + 6y) - 8x= 2x^{4}y^{2} + 11x^{3}y - y - 8x \]

Step 3: Determine the degree of the polynomial

In order to determine the degree of the resulting polynomial, identify the term with the highest power. In this case, the term with the highest power is \(2x^{4}y^{2}\), so the degree of the polynomial is 4 (which is the power of the variable x in this term).

Key Concepts

Combining Like Terms

When working with polynomials, combining like terms is a fundamental skill. It involves adding or subtracting variables that are the same, meaning they have identical variable parts raised to the same exponent. For example, in the expression \(5x^{4}y^{2} - 3x^{4}y^{2}\), both terms are 'like' because they contain the variable \(x\) to the fourth power and \(y\) to the second power. By combining them, we get \(2x^{4}y^{2}\).

To effectively combine like terms, simply add or subtract the coefficients (the numbers in front of the variables) and keep the variable part unchanged. Arranging terms so that like terms are next to each other can also simplify the process, enabling you to quickly identify and combine them. This is an essential step in many algebraic processes, including polynomial addition and subtraction.

Polynomial Subtraction

Regarding \(polynomial subtraction\), understanding that subtraction is essentially adding a negative is key. Let’s consider the given exercise where we have to subtract one polynomial from another. The first step is to change the sign of each term in the polynomial that is being subtracted.

When we perform the operation \( (5x^{4}y^{2}+6x^{3}y-7y)-(3x^{4}y^{2}-5x^{3}y-6y+8x) \), it becomes \( (5x^{4}y^{2}+6x^{3}y-7y) + (-3x^{4}y^{2}+5x^{3}y+6y-8x) \), making it easier to combine like terms. This changes the subtraction problem into an addition problem, which is typically more straightforward to solve.

Degree of Polynomial

The degree of a polynomial is a measure of the polynomial's 'size' in terms of its most significant terms' exponents. It is found by identifying the term with the highest combined exponent value. For example, in the term \(2x^{4}y^{2}\), the degree is 4 + 2 = 6, as we sum the exponents of \(x\) and \(y\). However, in the context of multiple variables, it's common to refer to the degree in terms of just one variable.

In the exercise, we look at the terms in the resulting polynomial to determine its degree. The highest degree term in the simplified polynomial is \(2x^{4}y^{2}\), which implies that the polynomial has a degree of 4 with respect to the variable \(x\) since that's the highest exponent of \(x\) present.

Polynomial Algebra

The practice of polynomial algebra encompasses the operations we perform with polynomials, including addition, subtraction, multiplication, and division, as well as factoring and finding roots of equations. In the context of our exercise, the steps taken to subtract one polynomial from another and combine like terms are part of polynomial algebra. It's crucial to understand how to rearrange terms, change signs for subtraction, and combine like terms to simplify and solve polynomial equations.

The skills applied in polynomial operations are foundational for studying higher-level algebra, calculus, and many practical applications in science and engineering.