Polynomial Sum Explained Step By Step Solution

This article will delve into the process of adding polynomials, providing a step-by-step explanation to solve the given problem. Polynomial addition is a fundamental concept in algebra, and mastering it is crucial for more advanced mathematical topics. We'll break down the problem, explain the underlying principles, and guide you through the solution. The question we aim to solve is: Which polynomial represents the sum of $(3x^2 + 7x + 3)$ and $(5x^2 + 12x)$?

Breaking Down the Basics of Polynomials

Before we tackle the problem directly, let's clarify what polynomials are and how they work. A polynomial is an expression consisting of variables (usually denoted by letters like x) and coefficients, combined using addition, subtraction, and non-negative integer exponents. Examples of polynomials include $x^2 + 2x + 1$, $3x^3 - 4x + 7$, and even simple expressions like $5x$ or $9$. The terms in a polynomial are the individual parts separated by addition or subtraction. For instance, in the polynomial $3x^2 + 7x + 3$, the terms are $3x^2$, $7x$, and $3$.

The degree of a polynomial is the highest power of the variable in the polynomial. In the example $3x^2 + 7x + 3$, the highest power of x is 2, so the degree of the polynomial is 2. Similarly, in the polynomial $5x^3 - 2x + 1$, the degree is 3 because the highest power of x is 3. Understanding the degree helps in classifying polynomials. A polynomial of degree 2 is called a quadratic, degree 3 is a cubic, and so on.

Adding Polynomials: The Core Principle

The key to adding polynomials lies in combining like terms. Like terms are terms that have the same variable raised to the same power. For example, $3x^2$ and $5x^2$ are like terms because both have x raised to the power of 2. Similarly, $7x$ and $12x$ are like terms because both have x raised to the power of 1 (which is usually not explicitly written). However, $3x^2$ and $7x$ are not like terms because the powers of x are different.

To add polynomials, you simply add the coefficients of the like terms. The variable and its exponent remain the same. For instance, to add $3x^2$ and $5x^2$, you add the coefficients 3 and 5, resulting in $8x^2$. The x² part stays the same. This principle is based on the distributive property of multiplication over addition. We can think of $3x^2 + 5x^2$ as $(3 + 5)x^2$, which simplifies to $8x^2$.

When adding multiple polynomials, it's helpful to align the like terms vertically. This makes it easier to see which terms need to be combined. If a term is missing in one of the polynomials, you can treat it as if it has a coefficient of 0. For example, if you're adding $x^2 + 3x + 2$ and $2x^3 + 5$, you can rewrite the second polynomial as $2x^3 + 0x^2 + 0x + 5$ to align the terms properly.

Step-by-Step Solution: Adding the Given Polynomials

Now, let's apply these principles to the problem at hand. We need to find the sum of the polynomials $3x^2 + 7x + 3$ and $5x^2 + 12x$. The first step is to align the like terms. We can write the problem in a vertical format as follows:

  3x^2 + 7x + 3
+ 5x^2 + 12x
-----------

Notice that the second polynomial, $5x^2 + 12x$, doesn't have a constant term (a term without any variable). We can think of it as having a constant term of 0, but we don't need to explicitly write it in this case.

Next, we add the coefficients of the like terms. Let's start with the $x^2$ terms. We have $3x^2$ and $5x^2$. Adding the coefficients 3 and 5, we get 8. So, the sum of the $x^2$ terms is $8x^2$.

Now, let's add the x terms. We have $7x$ and $12x$. Adding the coefficients 7 and 12, we get 19. So, the sum of the x terms is $19x$.

Finally, we add the constant terms. In the first polynomial, we have a constant term of 3. The second polynomial doesn't have a constant term, so we can think of it as having a constant term of 0. Adding 3 and 0, we get 3.

Combining these results, we get the sum of the polynomials as $8x^2 + 19x + 3$. Therefore, the polynomial that represents the sum of $3x^2 + 7x + 3$ and $5x^2 + 12x$ is $8x^2 + 19x + 3$.

Examining the Answer Choices

Now that we've found the sum, let's look at the answer choices provided:

A. $15x^2 + 19x + 3$ B. $8x^2 + 84x + 3$ C. $15x^2 + 84x + 3$ D. $8x^2 + 19x + 3$

Comparing our result, $8x^2 + 19x + 3$, with the answer choices, we see that option D, $8x^2 + 19x + 3$, is the correct answer. The other options have incorrect coefficients for the $x^2$ and x terms, indicating errors in the addition process.

Common Mistakes to Avoid in Polynomial Addition

While polynomial addition is relatively straightforward, there are some common mistakes that students often make. Being aware of these pitfalls can help you avoid them:

1. Combining Unlike Terms: One of the most frequent errors is adding terms that are not like terms. For example, adding $3x^2$ and $7x$ is incorrect because these terms have different powers of x. Remember, you can only add terms with the same variable and exponent.
2. Incorrectly Adding Coefficients: Another common mistake is adding the coefficients incorrectly. Double-check your arithmetic to ensure you're adding the numbers correctly. It's helpful to write out the addition steps clearly to avoid errors.
3. Forgetting to Include All Terms: When adding polynomials, make sure you include all the terms. If a polynomial is missing a term (e.g., the x term in $5x^2 + 3$), remember to account for it by treating it as having a coefficient of 0.
4. Sign Errors: Be careful with signs, especially when dealing with subtraction or negative coefficients. Ensure you're applying the correct sign to each term when adding or subtracting.
5. Not Aligning Like Terms: Failing to align like terms vertically can lead to confusion and errors. Taking the time to align the terms properly before adding can significantly reduce mistakes.

Practice Problems for Mastery

To solidify your understanding of polynomial addition, it's essential to practice. Here are a few additional problems you can try:

1. Add $(2x^3 - 5x^2 + x - 7)$ and $(x^3 + 3x^2 - 4x + 2)$.
2. Find the sum of $(4x^2 - 9x + 1)$ and $(-2x^2 + 6x - 5)$.
3. Combine $(7x^4 + 2x^2 - 3)$ and $(5x^3 - x^2 + 8)$.
4. What is the result of adding $(x^2 + 5x)$ and $(3x^2 - 2x + 4)$?
5. Simplify $(6x^3 - 4x + 9) + (2x^3 + 7x^2 - 5x)$.

By working through these problems, you'll gain confidence in your ability to add polynomials accurately and efficiently. Remember to focus on aligning like terms, adding coefficients correctly, and avoiding common mistakes.

Real-World Applications of Polynomial Addition

Polynomials are not just abstract mathematical concepts; they have numerous applications in various fields. Polynomial addition is used in engineering, physics, computer graphics, and economics, among others. Understanding how to manipulate polynomials is essential for solving real-world problems in these areas.

In engineering, polynomials are used to model curves and surfaces, calculate areas and volumes, and analyze systems. For example, engineers might use polynomials to describe the trajectory of a projectile or the shape of a bridge. Adding polynomials can help in combining different components of a system or calculating overall performance.

In physics, polynomials are used to describe motion, energy, and other physical quantities. For instance, the position of an object moving under constant acceleration can be described by a quadratic polynomial. Adding polynomials can be useful in analyzing the combined effects of multiple forces or motions.

Computer graphics relies heavily on polynomials for rendering images and creating animations. Curves and surfaces are often represented using polynomials, and polynomial operations are used to manipulate these shapes. Polynomial addition can be used to combine different graphical elements or to create smooth transitions between shapes.

In economics, polynomials can model cost, revenue, and profit functions. Analyzing these functions often involves polynomial operations, including addition. For example, adding two cost functions can give the total cost of producing multiple products.

Conclusion The Significance of Polynomial Addition

In conclusion, adding polynomials is a fundamental algebraic skill with wide-ranging applications. By understanding the basic principles of combining like terms and avoiding common mistakes, you can confidently solve polynomial addition problems. The solution to the given problem, which asks for the sum of $3x^2 + 7x + 3$ and $5x^2 + 12x$, is $8x^2 + 19x + 3$, corresponding to answer choice D. Mastering this skill is not only crucial for success in mathematics but also for various real-world applications in science, engineering, and other fields. So, keep practicing, and you'll become proficient in polynomial addition in no time!