Problem 62 - College Algebra

Question

Perform the indicated operations Indicate the degree of the resulting polynomial. $$ \left(7 x^{4} y^{2}-5 x^{2} y^{2}+3 x y\right)+\left(-18 x^{4} y^{2}-6 x^{2} y^{2}-x y\right) $$

Step-by-Step Solution

Answer

The simplified polynomial is \(-11 x^{4} y^{2} - 11 x^{2} y^{2} + 2 x y\). The degree of this polynomial is 6.

Step 1: Simplification of polynomials

Combine the like terms in the polynomials which includes \(7 x^{4} y^{2} -18 x^{4} y^{2}\), \(-5 x^{2} y^{2}-6 x^{2} y^{2}\) and \(3 x y-x y\). The steps to combine the like terms are as follows: \(7 x^{4} y^{2} - 18 x^{4} y^{2} = -11 x^{4} y^{2}\), \(-5 x^{2} y^{2} -6 x^{2} y^{2} = -11 x^{2} y^{2}\), \(3 x y - x y = 2 x y\).

Step 2: Writing the simplified polynomial

After adding the like terms, the polynomial is simplified to: \(-11 x^{4} y^{2} - 11 x^{2} y^{2} + 2 x y\).

Step 3: Determine the degree of the polynomial

The degree of a polynomial is determined by the highest power of the variable in the polynomial. Each term in the polynomial has degree determined by the sum of the exponents of its variables. The degree of \(-11 x^{4} y^{2}\) is 4+2=6, the degree of \(-11 x^{2} y^{2}\) is 2+2=4, and the degree of \(2 x y\) is 1+1=2. The highest of these degrees is 6, so the degree of the polynomial \(-11 x^{4} y^{2} - 11 x^{2} y^{2} + 2 x y\) is 6.

Key Concepts

Simplifying Polynomials

When working with polynomials, you're likely to encounter the process of simplification, which is an essential skill in algebra. Simplifying a polynomial involves reducing it to its simplest form. This does not mean solving a polynomial but rather consolidating and minimizing the number of terms whenever possible to make it easier to work with. In the context of the exercise given, simplification is achieved by combining like terms. A term is considered a 'like term' to another if they have the same variables raised to the same powers, even if their coefficients are different. Remember, the coefficients are the numerical parts of the term. You simply add or subtract the coefficients, and keep the variable part unchanged.
To illustrate, when you have two terms, such as \(7 x^{4} y^{2}\) and \( -18 x^{4} y^{2}\), they can be combined by subtracting their coefficients: \(7 - (-18)\) equals 25, so they simplify to \(25 x^{4} y^{2}\). For more complex polynomials, make sure to apply this technique methodically to each group of like terms.

Degree of a Polynomial

Understanding the degree of a polynomial is crucial for grasping the complexity of algebraic equations. The degree of a polynomial is determined by the term with the highest sum of exponents of its variables and gives us a lot of information about the polynomial, such as its possible number of zeros and its end behavior when graphed.
In the exercise provided, we encounter different terms within a polynomial. To find the degree of each term, simply add the exponents of the variables within that term. For example, the term \(x^{4}y^{2}\) has a degree of 6, because 4 (from \(x^{4}\)) plus 2 (from \(y^{2}\)) equals 6. Once you identify the degrees of all terms, the highest one represents the degree of the entire polynomial; this is important because it tells us the polynomial's most basic characteristics. In terms of polynomial hierarchy, the higher the degree, the more complex the polynomial.

Combining Like Terms

The concept of 'combining like terms' is a cornerstone of simplifying algebraic expressions and needs particular attention. Like terms are terms that have exactly the same variable components, regardless of their coefficients. To combine them, align the terms with similar variables and then add or subtract their coefficients as necessary.
Using our exercise as the basis, observe that \(7 x^{4} y^{2}\) and \( -18 x^{4} y^{2}\) are like terms, as they both contain the variables \(x\) and \(y\) raised to the fourth and second power respectively. Their coefficients, 7 and -18, are added together to get \( -11 x^{4} y^{2}\). Apply this technique across all terms with shared variables in a polynomial to effectively simplify the expression. This strategy streamlines the polynomial so it's easier to evaluate or manipulate later on in your calculations.