Problem 65 - Algebra and Trigonometry

Question

In Exercises 59–66, perform the indicated operations. Indicate the degree of the resulting polynomial. $$ \left(3 x^{4} y^{2}+5 x^{3} y-3 y\right)-\left(2 x^{4} y^{2}-3 x^{3} y-4 y+6 x\right) $$

Step-by-Step Solution

Answer

The resultant polynomial after subtraction is \(x^{4}y^{2}+8x^{3}y+y-6x\) and its degree is 4.

Step 1: Identify like terms

Like terms in polynomials are those terms which have the same variables and powers. Here the like terms are \(3 x^{4} y^{2}\) and \(2 x^{4} y^{2}\), \(5 x^{3 }y\) and \(-3 x^{3} y\), \(-3 y\) and \(-4 y\)

Step 2: Subtract like terms

Once we’ve matched up the like terms between the two polynomials, we can subtract one term from another. This gives us \((3 x^{4} y^{2}-2 x^{4} y^{2})+(5 x^{3} y+3 x^{3} y)+(-3 y+4 y)-6x\).

Step 3: Simplify each pair of like terms

By doing the subtraction we get \(x^{4}y^{2}+8x^{3}y+y-6x\)

Step 4: Determine the degree of the polynomial

Degree of a term in a polynomial is the exponent of its variable. The degree of a polynomial is the highest degree of any term within the polynomial. Here, the degree is 4, as \(x^{4}y^{2}\) has the highest degree.

Key Concepts

Like Terms

In polynomial subtraction, one of the first steps is to identify "like terms". Like terms are terms that have the same variables raised to the same powers. If you're handling expressions with multiple variables and exponents, it’s crucial to match these exactly.
For example, in the expression \(3x^{4}y^{2}\), the variables are \(x\) and \(y\), raised to the 4th and 2nd powers respectively. Any other term needs to have these exact same variables with the same exponents to be considered like terms.

• Example of like terms: \(3x^{4}y^{2}\) and \(2x^{4}y^{2}\)
• Non-like terms: \(3x^{4}y^{2}\) and \(3x^{4}y\) because the power of \(y\) is different.

Recognizing like terms allows you to combine them into a simpler form during subtraction or addition by directly executing mathematical operations just on their coefficients.

Degree of a Polynomial

The "degree of a polynomial" is another important aspect to consider. The degree is defined as the greatest degree among its terms after simplifying the expression.
Each term’s degree is the sum of the exponents of its variables. For instance, in the term \(x^{4}y^{2}\), the degree is \(4 + 2 = 6\), because it comprises \(x\) to the 4th power and \(y\) to the 2nd.

• A constant term (such as 5) has a degree of 0 because there is no variable with an exponent.
• A single variable term, like \(-6x\), has a degree of 1.

After performing polynomial operations, the term with the highest degree dictates the degree of the entire polynomial. This helps categorize polynomials and understand their complexity and behavior in equations.

Simplifying Polynomials

"Simplifying polynomials" involves reducing polynomials to their simplest forms. This often means combining like terms, removing unnecessary parentheses, and making calculations to shorten the expression.
The steps to simplifying a polynomial tend to involve:

• Identifying like terms across the entire polynomial expression.
• Performing arithmetic operations (addition, subtraction) to combine these terms.
• Rewriting the simplified expression in a standard format, usually in descending order of degree.

This process is fundamental to solving polynomial equations effectively. For example, in the expression \((3x^{4}y^{2} - 2x^{4}y^{2}) + (-3y + 4y) - 6x\), after simplification, it becomes \(x^{4}y^{2} + 8x^{3}y + y - 6x\). By simplifying, we ensure the expression is more manageable and ready to work with for further mathematical operations or problem-solving.