The Sum Of Two Polynomials Is $8d^5 - 3c^3d^2 + 5c^2d^3 - 4cd^4 + 9$. If One Addend Is $2d^5 - C^3d^2 + 8cd^4 + 1$, What Is The Other Addend?A. $6d^5 - 2c^3d^2 + 5c^2d^3 - 12cd^4 + 8$ B. $6d^5 - 4c^3d^2 + 3c^2d^3 - 4cd^4

Introduction

In algebra, polynomials are expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. When we add two polynomials, we combine like terms to simplify the resulting expression. In this article, we will explore how to find the missing addend in a given polynomial sum.

Understanding Polynomial Addition

Polynomial addition is a straightforward process. When adding two polynomials, we combine like terms by adding their coefficients. For example, if we have two polynomials:

• $2x^2 + 3x - 4$
• $x^2 + 2x + 5$

We can add them by combining like terms:

• $(2x^2 + 3x - 4) + (x^2 + 2x + 5)$
• $= 2x^2 + x^2 + 3x + 2x - 4 + 5$
• $= 3x^2 + 5x + 1$

The Given Problem

We are given the sum of two polynomials:

$8d^5 - 3c^3d^2 + 5c^2d^3 - 4cd^4 + 9$

One of the addends is:

$2d^5 - c^3d^2 + 8cd^4 + 1$

Our goal is to find the other addend.

Step 1: Identify Like Terms

To find the missing addend, we need to identify like terms in the given sum. Like terms are terms with the same variable and exponent. In this case, we have the following like terms:

• $d^5$ terms: $8d^5$ and $2d^5$
• $c^3d^2$ terms: $-3c^3d^2$ and $-c^3d^2$
• $c^2d^3$ terms: $5c^2d^3$ (no other term with this variable and exponent)
• $cd^4$ terms: $-4cd^4$ and $8cd^4$
• Constant terms: $9$ and $1$

Step 2: Combine Like Terms

Now that we have identified like terms, we can combine them by adding their coefficients. Let's do this for each set of like terms:

• $d^5$ terms: $8d^5 + 2d^5 = 10d^5$
• $c^3d^2$ terms: $-3c^3d^2 - c^3d^2 = -4c^3d^2$
• $c^2d^3$ terms: $5c^2d^3$ (no other term with this variable and exponent)
• $cd^4$ terms: $-4cd^4 + 8cd^4 = 4cd^4$
• Constant terms: $9 + 1 = 10$

Step 3: Write the Missing Addend

Now that we have combined like terms, we can write the missing addend by subtracting the given addend from the sum. Let's do this:

• $(8d^5 - 3c^3d^2 + 5c^2d^3 - 4cd^4 + 9) - (2d^5 - c^3d^2 + 8cd^4 + 1)$
• $= 8d^5 - 2d^5 - 3c^3d^2 + c^3d^2 + 5c^2d^3 - 4cd^4 - 8cd^4 + 9 - 1$
• $= 6d^5 - 4c^3d^2 + 5c^2d^3 - 12cd^4 + 8$

Conclusion

In this article, we have explored how to find the missing addend in a given polynomial sum. We identified like terms, combined them by adding their coefficients, and wrote the missing addend by subtracting the given addend from the sum. The missing addend is:

$6d^5 - 4c^3d^2 + 5c^2d^3 - 12cd^4 + 8$

This is the correct answer.

Answer

The correct answer is:

• A. $6d^5 - 4c^3d^2 + 5c^2d^3 - 12cd^4 + 8$

Q: What is the difference between a polynomial and an expression?

A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. An expression, on the other hand, can be any combination of variables, coefficients, and mathematical operations.

Q: How do you add two polynomials?

A: To add two polynomials, you combine like terms by adding their coefficients. Like terms are terms with the same variable and exponent.

Q: What are like terms?

A: Like terms are terms with the same variable and exponent. For example, $2x^2$ and $3x^2$ are like terms because they both have the variable $x$ and the exponent $2$.

Q: How do you combine like terms?

A: To combine like terms, you add their coefficients. For example, if you have the terms $2x^2$ and $3x^2$, you can combine them by adding their coefficients: $2x^2 + 3x^2 = 5x^2$.

Q: What is the missing addend in the given problem?

A: The missing addend is $6d^5 - 4c^3d^2 + 5c^2d^3 - 12cd^4 + 8$.

Q: How do you find the missing addend?

A: To find the missing addend, you need to identify like terms in the given sum, combine them by adding their coefficients, and then subtract the given addend from the sum.

Q: What is the importance of identifying like terms?

A: Identifying like terms is crucial in adding polynomials because it allows you to combine terms with the same variable and exponent, simplifying the resulting expression.

Q: Can you give an example of a polynomial sum?

A: Here's an example:

$(2x^2 + 3x - 4) + (x^2 + 2x + 5)$

To add these two polynomials, you need to combine like terms:

$(2x^2 + 3x - 4) + (x^2 + 2x + 5)$ $= 2x^2 + x^2 + 3x + 2x - 4 + 5$ $= 3x^2 + 5x + 1$

Q: What is the final answer to the given problem?

A: The final answer is $6d^5 - 4c^3d^2 + 5c^2d^3 - 12cd^4 + 8$.

Q: Why is it important to understand polynomial addition?

A: Understanding polynomial addition is crucial in algebra because it allows you to simplify complex expressions and solve equations involving polynomials.

Q: Can you give a real-world example of polynomial addition?

A: Here's a real-world example:

Imagine you have two boxes of apples, one containing 2 pounds of apples and the other containing 3 pounds of apples. If you combine the two boxes, you will have a total of 5 pounds of apples. This is an example of polynomial addition, where the two polynomials represent the weights of the apples in each box.

Conclusion

In this article, we have answered some common questions about polynomial addition, including what like terms are, how to combine them, and how to find the missing addend in a given polynomial sum. We have also provided examples and real-world applications to illustrate the importance of understanding polynomial addition.