Problem 65 - College Algebra

Question

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 degree of the polynomial is 6.

Step 1: Opposite Of The Second Polynomial

We will first remove the brackets from the second polynomial, remembering to change the sign of every term. So the equation becomes:\(3 x^{4} y^{2}+5 x^{3} y-3 y-2 x^{4} y^{2}+3 x^{3} y+4 y-6 x\)

Step 2: Combine Similar Terms

Next, we will combine the similar terms i.e., the terms with the same variables and the same powers. This results in:\((3x^{4} y^{2}-2x^{4} y^{2}) + (5x^{3} y+3x^{3} y) + (-3y+4y) - 6x\)This simplifies to:\(x^{4} y^{2} + 8x^{3} y + y - 6x\)

Step 3: Determine the Degree

The degree of the polynomial is the greatest sum of the exponents of its terms. In our polynomial, \(x^{4} y^{2}\) has the highest sum of exponents, 4+2=6. So, the degree of the polynomial is 6

Key Concepts

Degree of a Polynomial

Understanding the degree of a polynomial is crucial for grasping its behavior and characteristics. The degree of a polynomial is determined by the term with the highest sum of exponents. For instance, in our example, the polynomial becomes \(x^{4}y^{2} + 8x^{3}y + y - 6x\). Here, the term \(x^{4}y^{2}\) boasts the highest degree because it has the sum of exponents \(4 + 2 = 6\).
This is how we find out that the degree of our polynomial is 6. Remember that each variable in a term contributes its exponent to the total sum for that term. Thus, identifying the term with the largest sum gives us the polynomial's degree. Recognizing the degree helps in predicting how the function behaves as variables grow larger, as higher-degree polynomials can display more complex curves and changes.

Combining Like Terms

When working with polynomials, combining like terms is an essential step. Like terms are terms that have identical variable parts—meaning they have the same variables raised to the same powers.
In the provided exercise, after removing the parentheses and changing signs, we had the expression \(3x^{4}y^{2} + 5x^{3}y - 3y - 2x^{4}y^{2} + 3x^{3}y + 4y - 6x\).
To simplify, we identified and grouped like terms:

• Terms like \(3x^{4}y^{2}\) and \(-2x^{4}y^{2}\) were combined as \((3-2)x^{4}y^{2}\).
• Terms like \(5x^{3}y\) and \(3x^{3}y\) were combined as \((5+3)x^{3}y\).
• \(-3y\) and \(4y\) were combined as \((-3+4)y\).
• The \(-6x\) stood alone as there were no other terms with \(x\).

After combining, we obtain \(x^{4}y^{2} + 8x^{3}y + y - 6x\). Combining like terms makes the polynomial simpler and easier to handle for further operations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that contain numbers, variables, and operations. They form the base of algebra and can represent real-world situations and complex relationships.
In an algebraic expression, components combine through operations like addition, subtraction, multiplication, and division. Variables in expressions allow for flexibility and generalization. For example, the expression \(3x^{4}y^{2} + 5x^{3}y - 3y\) represents a polynomial which changes with different values of \(x\) and \(y\).
Understanding the components of an algebraic expression is essential for simplifying and solving equations. Correctly identifying coefficients (numerical multipliers of terms) and constants (terms without variables, though absent in this example) aids in manipulating expressions. Recognizing these elements is key for any further operations like solving equations or modeling scenarios.