Problem 60 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The resulting polynomial is \(2x^{2}y + 8xy\) and its degree is 3

Step 1: Combine like terms

Combine like terms that have the same variables and powers. This means adding/subtracting the coefficients of these terms. So, \(-2x^{2}y + x y + 4x^{2}y + 7xy\) can be rewritten as \((-2 + 4)x^{2}y + (1+7)xy \) which simplifies to \(2x^{2}y + 8xy\)

Step 2: Determine degree of the polynomial

The degree of a polynomial is the highest power of its variable. Here, the term with the highest degree is \(2x^{2}y\) and the degree is 2 + 1 (since \( y = y^{1} \)). So the degree of the resulting polynomial is 3

Key Concepts

Understanding the Degree of a Polynomial

The "degree of a polynomial" refers to the term with the highest total power of variables when you add their exponents together. For instance, in the polynomial \(2x^{2}y + 8xy\), we look at each term:

• \(2x^{2}y\) has exponents 2 for \(x\) and 1 for \(y\). Adding these gives a degree of 3.
• \(8xy\) has exponents 1 for \(x\) and 1 for \(y\), totaling 2.

The degree of the polynomial is the highest of these values, which is 3 in this case. This step is crucial as the degree gives us insight into the behavior and potential graph-related properties of the polynomial.

Combining Like Terms in Polynomials

"Combining like terms" involves simplifying expressions by merging terms with the same variables and exponents. Let's see how it's done with \(-2x^{2}y + xy + 4x^{2}y + 7xy\):

• Identify like terms. The like terms here are \(-2x^{2}y\) and \(4x^{2}y\), both sharing \(x^{2}y\).
• The terms \(xy\) and \(7xy\) are also like terms, sharing \(xy\).
• Combine by adding or subtracting their coefficients. \(-2 + 4 = 2\) for \(x^{2}y\) and \(1 + 7 = 8\) for \(xy\).

This results in the simplified polynomial \(2x^{2}y + 8xy\). Understanding this process is key in algebra to create simpler, more workable expressions.

Polynomial Addition Explained

"Polynomial addition" might sound complex, but it simply involves adding matching terms together. In our problem, we had two polynomials: \((-2 x^{2} y + xy)\) and \((4 x^{2} y + 7xy)\).

• Begin by rewriting the expression with "like terms" next to each other: \(-2x^{2}y + 4x^{2}y + xy + 7xy\).
• Add the coefficients of matching terms: for \(x^{2}y\) terms, add \(-2\) and \(4\), and for \(xy\) terms, add \(1\) and \(7\).
• This gives the result \(2x^{2}y + 8xy\), making it clear and concise.

Mastering polynomial addition allows us to manipulate and simplify complex algebraic expressions effectively.