Polynomial In Standard Form Combining Like Terms

Understanding Polynomials and Standard Form

When working with polynomials, it's crucial to understand how to combine like terms and express the polynomial in standard form. Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication. The degree of a term in a polynomial is the sum of the exponents of the variables in that term. Like terms are terms that have the same variables raised to the same powers. Combining like terms involves adding or subtracting the coefficients of these terms.

The standard form of a polynomial is a way of writing it in descending order of the degrees of its terms. This means the term with the highest degree comes first, followed by the term with the next highest degree, and so on, until the term with the lowest degree (the constant term) is written last. Expressing a polynomial in standard form makes it easier to compare polynomials, perform operations on them, and analyze their behavior. The standard form also provides a consistent way to represent polynomials, which is essential in various mathematical applications.

Before diving into the specific problem, let's solidify our understanding with an example. Consider the polynomial $3x^2 + 2x - 5 + x^2 - 4x + 2$. To express this in standard form, we first identify the like terms: $3x^2$ and $x^2$ are like terms, and $2x$ and $-4x$ are like terms. Combining these, we get $(3x^2 + x^2) + (2x - 4x) + (-5 + 2)$, which simplifies to $4x^2 - 2x - 3$. This is the standard form of the polynomial, with the terms arranged in descending order of their degrees (2, 1, and 0).

Problem Breakdown and Solution Strategy

Now, let's tackle the given polynomial: $8mn^5 - 2m^6 + 5m^2n^4 - m^3n^3 + n^6 - 4m^6 + 9m^2n^4 - mn^5 - 4m^2n^3$. Our goal is to combine the like terms and express the result in standard form. This involves several steps:

1. Identify Like Terms: Look for terms with the same variables raised to the same powers. For example, $8mn^5$ and $-mn^5$ are like terms, as are $5m^2n^4$ and $9m^2n^4$. Similarly, $-2m^6$ and $-4m^6$ are like terms, and so on.
2. Combine Like Terms: Add or subtract the coefficients of the like terms. For instance, $8mn^5 - mn^5 = 7mn^5$, and $5m^2n^4 + 9m^2n^4 = 14m^2n^4$.
3. Determine the Degree of Each Term: The degree of a term is the sum of the exponents of the variables. For example, the degree of $mn^5$ is $1 + 5 = 6$, the degree of $m^6$ is 6, the degree of $m^2n^4$ is $2 + 4 = 6$, and so on.
4. Arrange Terms in Descending Order of Degree: Write the terms in the order of their degrees, from highest to lowest. If terms have the same degree, they can be arranged alphabetically based on the variables.

Let's apply these steps to the given polynomial.

Step-by-Step Solution

1. Identify Like Terms: In the given polynomial $8mn^5 - 2m^6 + 5m^2n^4 - m^3n^3 + n^6 - 4m^6 + 9m^2n^4 - mn^5 - 4m^2n^3$, the like terms are:
   • $8mn^5$ and $-mn^5$
   • $-2m^6$ and $-4m^6$
   • $5m^2n^4$ and $9m^2n^4$
   • $-4m^2n^3$ has no other like terms
   • $-m^3n^3$ has no other like terms
   • $n^6$ has no other like terms
2. Combine Like Terms: Combine the coefficients of the like terms:
   • $8mn^5 - mn^5 = 7mn^5$
   • $-2m^6 - 4m^6 = -6m^6$
   • $5m^2n^4 + 9m^2n^4 = 14m^2n^4$ So, we have: $7mn^5 - 6m^6 + 14m^2n^4 - m^3n^3 - 4m^2n^3 + n^6$
3. Determine the Degree of Each Term: Calculate the degree of each term:
   • $7mn^5$: Degree is $1 + 5 = 6$
   • $-6m^6$: Degree is 6
   • $14m^2n^4$: Degree is $2 + 4 = 6$
   • $-m^3n^3$: Degree is $3 + 3 = 6$
   • $-4m^2n^3$: Degree is $2 + 3 = 5$
   • $n^6$: Degree is 6
4. Arrange Terms in Descending Order of Degree: Arrange the terms in descending order of their degrees. Since multiple terms have a degree of 6, we can arrange them alphabetically based on the variables:
   • $n^6$ (Degree 6)
   • $-6m^6$ (Degree 6)
   • $7mn^5$ (Degree 6)
   • $14m^2n^4$ (Degree 6)
   • $-m^3n^3$ (Degree 6)
   • $-4m^2n^3$ (Degree 5)

Final Result in Standard Form

Combining the results, the polynomial in standard form is:

$n^6 - 6m^6 + 7mn^5 + 14m^2n^4 - m^3n^3 - 4m^2n^3$

Notice that we have a term of degree 5 while all other terms are of degree 6. It appears there was a mistake in the original polynomial or the provided answer choices, as this expression cannot be further simplified in a typical sense. However, if we were to strictly follow the steps, we would present the polynomial as:

$n^6 - 6m^6 + 7mn^5 + 14m^2n^4 - m^3n^3 - 4m^2n^3$

If the intention was for all terms to have degree 6 or to have like terms that could fully combine, there might be an error in the original question. Nonetheless, following the procedure for combining like terms and ordering by degree, this is the resultant standard form.

Conclusion

In this detailed walkthrough, we methodically combined like terms and expressed the given polynomial in standard form. Understanding these concepts is essential for advanced algebraic manipulations and problem-solving. By breaking down the problem into manageable steps, we ensured clarity and accuracy in our solution. Remember, identifying like terms, combining them, determining degrees, and arranging terms in descending order are the key elements of expressing a polynomial in standard form. This systematic approach not only helps in solving the problem accurately but also enhances understanding and application in various mathematical contexts.