Polynomial Simplification Identifying Terms And Degree

In the realm of algebra, polynomials stand as fundamental expressions, playing a crucial role in various mathematical and scientific applications. A polynomial is essentially an expression comprising variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Simplifying polynomials is a critical skill in algebra, allowing us to more easily understand and manipulate these expressions. This article delves into the process of simplifying a polynomial and identifying its key characteristics, such as the number of terms and the degree.

Simplifying Polynomials: A Step-by-Step Approach

To simplify a polynomial, we need to combine like terms. Like terms are terms that have the same variables raised to the same powers. For instance, $3x^2$ and $-5x^2$ are like terms because they both have the variable x raised to the power of 2. However, $3x^2$ and $3x^3$ are not like terms because the exponents of x are different. Similarly, $2xy$ and $-4xy$ are like terms, but $2xy$ and $2x$ are not like terms because one term has both x and y, while the other has only x.

To simplify a polynomial, we follow these steps:

1. Identify like terms: Look for terms in the polynomial that have the same variables raised to the same powers.
2. Combine like terms: Add or subtract the coefficients of the like terms. Remember that when adding or subtracting like terms, we only combine the coefficients; the variables and their exponents remain the same.

Let's illustrate this with an example. Consider the polynomial:

$$5x^3 + 2x^2 - 3x + 7 - 2x^3 + x^2 + 5x - 3$$

First, identify the like terms:

• $5x^3$ and $-2x^3$ are like terms.
• $2x^2$ and $x^2$ are like terms.
• $-3x$ and $5x$ are like terms.
• $7$ and $-3$ are like terms (these are constant terms).

Next, combine the like terms:

• $5x^3 - 2x^3 = 3x^3$
• $2x^2 + x^2 = 3x^2$
• $-3x + 5x = 2x$
• $7 - 3 = 4$

So, the simplified polynomial is:

$$3x^3 + 3x^2 + 2x + 4$$

This simplified form is much easier to work with than the original polynomial. It has fewer terms, making it simpler to evaluate, differentiate, or integrate.

Applying Simplification to a Specific Polynomial

Now, let's apply this simplification process to the polynomial given in the problem:

$$-3x^4y^3 + 8xy^5 - 3 + 18x^3y^4 - 3xy^5$$

1. Identify like terms: In this polynomial, the like terms are $8xy^5$ and $-3xy^5$. The other terms, $-3x^4y^3$, $-3$, and $18x^3y^4$, do not have any like terms to combine with.

2. Combine like terms: Combine $8xy^5$ and $-3xy^5$:

   $$8xy^5 - 3xy^5 = 5xy^5$$

After combining the like terms, the simplified polynomial is:

$$-3x^4y^3 + 5xy^5 + 18x^3y^4 - 3$$

Now that we have simplified the polynomial, we can determine its characteristics, such as the number of terms and the degree.

Determining the Number of Terms

The terms of a polynomial are the individual expressions separated by addition or subtraction signs. In the simplified polynomial:

$$-3x^4y^3 + 5xy^5 + 18x^3y^4 - 3$$

We can identify four distinct terms:

1. $-3x^4y^3$
2. $5xy^5$
3. $18x^3y^4$
4. $-3$

Therefore, the simplified polynomial has 4 terms.

Determining the Degree of a Polynomial

The degree of a polynomial is the highest sum of the exponents of the variables in any term of the polynomial. To find the degree, we need to consider each term separately:

1. For the term $-3x^4y^3$, the exponents are 4 and 3. The sum of the exponents is $4 + 3 = 7$.
2. For the term $5xy^5$, the exponents are 1 (for x) and 5 (for y). The sum of the exponents is $1 + 5 = 6$.
3. For the term $18x^3y^4$, the exponents are 3 and 4. The sum of the exponents is $3 + 4 = 7$.
4. For the constant term $-3$, the degree is 0 because there are no variables.

Now, we compare the sums of the exponents for each term: 7, 6, 7, and 0. The highest sum is 7. Therefore, the degree of the polynomial is 7.

Analyzing the Answer Choices

Now that we have simplified the polynomial and determined its characteristics, we can analyze the answer choices provided in the original problem.

The simplified polynomial is:

$$-3x^4y^3 + 5xy^5 + 18x^3y^4 - 3$$

We found that it has 4 terms and a degree of 7.

Let's look at the answer choices:

A. It has 3 terms and a degree of 5. B. It has 3 terms and a degree of 7. C. It has 4 terms and a degree of 5. D. It has 4 terms and a degree of 7.

Comparing our findings with the answer choices, we can see that the correct answer is:

D. It has 4 terms and a degree of 7.

Conclusion

Simplifying polynomials is a fundamental skill in algebra. By combining like terms, we can reduce the complexity of an expression and make it easier to analyze. In the case of the polynomial $-3x^4y^3 + 8xy^5 - 3 + 18x^3y^4 - 3xy^5$, we simplified it to $-3x^4y^3 + 5xy^5 + 18x^3y^4 - 3$. This simplified form allowed us to easily identify that the polynomial has 4 terms and a degree of 7. Understanding these characteristics is crucial for further algebraic manipulations and applications of polynomials in various fields of mathematics and science. The degree of a polynomial, in particular, provides valuable information about its behavior and properties, especially when dealing with polynomial functions and their graphs. For instance, the degree of a polynomial function can indicate the maximum number of roots it may have, as well as the end behavior of its graph. Similarly, the number of terms can give insights into the complexity of the polynomial and the potential for factoring or simplifying it further. In summary, mastering the art of simplifying polynomials and identifying their characteristics is an essential step in developing a strong foundation in algebra and its applications.