Problem 66 - College Algebra

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 operations is \(2 x^{4} y^{2} + 11 x^{3} y - 13 y + 8 x\) and its degree is 6.

Step 1: Distribute the Negative Sign

Always remember that when you are subtracting polynomials, you are actually adding the opposite. So distribute the negative sign to each term in the second parentheses: \(5 x^{4} y^{2}+6 x^{3} y-7 y- (3 x^{4} y^{2}-5 x^{3} y+6 y-8 x)\) becomes \(5 x^{4} y^{2}+6 x^{3} y-7 y -3 x^{4} y^{2}+5 x^{3} y-6 y+8 x\).

Step 2: Combine Like Terms

Next, rearrange and combine the like terms in the polynomial: This gives us \( (5 x^{4} y^{2} - 3 x^{4} y^{2}) + (6 x^{3} y + 5 x^{3} y) - (7 y + 6 y) + 8 x\), which simplifies to \(2 x^{4} y^{2} + 11 x^{3} y - 13 y + 8 x \).

Step 3: Identify the Degree of the Polynomial

The degree of a polynomial is the highest power of its variable(s). Looking at each term of the resulting polynomial, the first term, \( 2 x^{4} y^{2} \), has a degree of 6 (4 from \(x\) and 2 from \(y\)), the second term, \( 11 x^{3} y \), has a degree of 4, the third term \( -13 y \) has a degree of 1 and the final term, \(8 x\), has a degree of 1. Thus, the degree of the resulting polynomial is 6.

Key Concepts

Degree of Polynomial

In the world of algebra, a polynomial's degree is one of its defining traits. It is essentially the highest degree of its terms when the polynomial is expressed in its standard form. Each term's degree is determined by adding up the exponents of the variables within it. For example, if you have a term like \( x^3y^2 \), the degree of this term is 5, as you add 3 (from \( x^3 \)) and 2 (from \( y^2 \)). The degree gives us vital insights about the polynomial's behavior and its potential graph shape.

When evaluating a polynomial, always identify each term's degree first. The term with the largest total gives the overall degree of the polynomial. Understanding this concept is crucial when solving problems, as it can impact both the solution method and interpretation of results.

Combining Like Terms

Combining like terms is a fundamental step in simplifying polynomials. Like terms are terms that contain the same variable raised to the same power. For instance, \( 3x^2 \) and \( 5x^2 \) are like terms, but \( 3x^2 \) and \( 4x^3 \) are not.

To combine them, you add or subtract the coefficients while keeping the variable part unchanged. For instance, in the expression \( 6x^3y + 5x^3y \), these are like terms because they have the same variables with the same powers. Thus, you can add the coefficients, obtaining \( 11x^3y \).

Mastering the skill of identifying and combining like terms allows you to simplify expressions and solve equations more efficiently. It reduces complexity and makes further operations or extractions of information, like the polynomial's degree, more manageable.

Distributing Negative Sign

Distributing a negative sign is often required when subtracting polynomials. This task should not be overlooked, as improper distribution can lead to errors. Distributing ensures that every term within the parentheses changes its sign before combining with terms outside.

Say you have an expression such as \(- (3x^4y^2 - 5x^3y + 6y - 8x)\), the negative sign means each term inside becomes the opposite: \(-3x^4y^2 + 5x^3y - 6y + 8x\). Neglecting this important step could result in incorrect solutions and complications in further simplification steps.

Get into the habit of checking each term thoroughly to prevent any sign errors. This check assures that further calculations, like combining like terms, proceed smoothly and accurately.

Polynomial Degree Determination

Finding the degree of a polynomial is crucial for understanding its behavior and characteristics. This is done by examining each term and identifying which has the highest degree. To do this efficiently, calculate the degree of each term and compare.

For instance, a polynomial such as \( 2x^4y^2 + 11x^3y - 13y + 8x \) requires evaluating each term. The highest degree comes from \( 2x^4y^2 \), where the total is 6 (4 from \( x \) and 2 from \( y \)).

Always ensure each term is analyzed, especially in multi-variable polynomials, as different combinations could change the hierarchy. This understanding is significant when predicting how a polynomial can interact in equations and real-world applications.