Problem 63 - College Algebra

Question

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

Step-by-Step Solution

Answer

The result of the operations on the given polynomials is \(7x^{3} + 6xy + y^{2}\), and its degree is 3.

Step 1: Distribute the Minus Sign

This will be done to the second polynomial. It changes the signs of each term in the second polynomial. \[-\left(6x^{3}-xy+4y^{2}\right) = -6x^{3} + xy - 4y^{2}\]

Step 2: Perform the Subtraction Operation

The similar terms will be subtracted from each other: \[\left(x^{3}+7xy-5y^{2}\right) - \left(-6x^{3}+xy-4y^{2}\right) = x^{3} -(-6x^{3}) + 7xy - xy + -5y^{2} -(-4y^{2})\] Simplifying this gives: \(7x^{3} + 6xy + y^{2}\)

Step 3: Identify the Degree of the Polynomial

The degree of a polynomial is determined by the largest sum of the exponents in any term of the polynomial. Looking at each term in the expression, the degrees are 3, 2 and 2. So, the degree of \(7x^{3} + 6xy + y^{2}\) is 3.

Key Concepts

Degree of a Polynomial

The degree of a polynomial is a fundamental concept in understanding how polynomials behave. It represents the highest sum of exponents in any single term of a polynomial. This is crucial because it tells us a lot about the polynomial's behavior, especially as the values of the variables increase.
To determine the degree:

• Look at each term separately.
• For each term, add up the exponents of all the variables.
• The highest sum you find is the degree of the polynomial.

In the resulting polynomial from our exercise, which is expressed as \(7x^{3} + 6xy + y^{2}\), identify each term's degree separately:

• The term \(7x^{3}\) has a degree of 3, since it is \(x^{3} \).
• The term \(6xy\) has a degree of 2, where both \(x\) and \(y\) are raised to the first power, totaling 2.
• The term \(y^{2}\) also has a degree of 2.

Since 3 is the highest degree among these terms, the polynomial's overall degree is 3. Understanding this concept helps in predicting how the polynomial graphically behaves and its end behavior.

Distributing Signs

Distributing signs correctly in mathematical operations ensures that terms are added and subtracted accurately, which is essential in polynomial operations. Distributing a sign, especially a negative, changes the operation for each term it affects.
When you have a negative sign in front of a polynomial, apply it to each term within the parentheses:

• Multiply each term by -1.
• This effectively flips the sign of each term.

For example, to distribute the negative sign in the expression \(-\left(6x^{3} - xy + 4y^{2}\right)\):

• The term \(6x^{3}\) becomes \(-6x^{3}\).
• The term \(-xy\) becomes \(+xy\).
• The term \(4y^{2}\) becomes \(-4y^{2}\).

This step is critical for ensuring any subtraction between polynomials is calculated correctly while avoiding unnecessary errors.

Subtraction of Polynomials

Subtraction of polynomials might sound daunting at first, but it's merely about organizing like terms and performing straightforward arithmetic. The process involves:

• Writing out both polynomials clearly.
• Distributing any negative signs as previously discussed.
• Aligning like terms (terms that have the same variables raised to the same power) from both expressions.

In our exercise, the initial subtraction setup was \(\left(x^{3}+7xy-5y^{2}\right) - \left(6x^{3}-xy+4y^{2}\right)\). After distributing the negative sign, it becomes easier to combine like terms:

• Combine the \(x^{3}\) terms: \(x^{3} - 6x^{3} = 7x^{3}\).
• Combine the \(xy\) terms: \(7xy + xy = 6xy\).
• Combine the \(y^{2}\) terms: \(-5y^{2} - 4y^{2} = y^{2}\).

The result is \(7x^{3} + 6xy + y^{2}\). Subtracting polynomials simplifies to adding their negatives, aligning terms, and performing simple arithmetic—making these problems approachable with practice.