Simplifying Polynomials A B - C In Simplest Form

Jul 13, 2025 by ADMIN 49 views

Polynomial Operations A B - C in Simplest Form

In mathematics, polynomials are fundamental algebraic expressions that consist of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Polynomial operations, such as addition, subtraction, and multiplication, are essential skills in algebra and are used extensively in various mathematical and scientific fields. This article delves into the process of simplifying polynomial expressions, specifically focusing on the operation $AB - C$ , where $A$ , $B$ , and $C$ are given polynomials. We will explore the steps involved in performing this operation, including substitution, multiplication, and combining like terms, to arrive at the simplest form of the resulting polynomial. Understanding polynomial operations is crucial for solving algebraic equations, graphing functions, and modeling real-world phenomena. This article aims to provide a comprehensive guide to simplifying polynomial expressions, equipping readers with the knowledge and skills to tackle similar problems with confidence.

Understanding Polynomials

Before we dive into the specific problem of simplifying $AB - C$ , it is essential to have a solid understanding of what polynomials are and the basic operations that can be performed on them. Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. A polynomial in one variable, say $x$ , can be written in the general form:

a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0

where $a_n, a_{n-1}, ..., a_1, a_0$ are the coefficients (constants) and $n$ is a non-negative integer representing the degree of the polynomial. Each term in the polynomial is a monomial, which is a product of a constant and a variable raised to a non-negative integer power. For example, $3x^2$ , $-5x$ , and $7$ are all monomials.

The degree of a polynomial is the highest power of the variable in the polynomial. For instance, the degree of the polynomial $3x^4 - 2x^2 + 5x - 1$ is 4. Polynomials can be classified based on their degree: a constant polynomial has degree 0, a linear polynomial has degree 1, a quadratic polynomial has degree 2, and so on.

Polynomial operations include addition, subtraction, multiplication, and division. When adding or subtracting polynomials, we combine like terms, which are terms with the same variable raised to the same power. For example, to add the polynomials $2x^2 + 3x - 1$ and $x^2 - 2x + 4$ , we combine the $x^2$ terms, the $x$ terms, and the constant terms:

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

When multiplying polynomials, we use the distributive property to multiply each term in one polynomial by each term in the other polynomial. For example, to multiply the polynomials $(x + 2)$ and $(x - 3)$ , we have:

(x + 2)(x - 3) = x(x - 3) + 2(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6

Polynomial division is a more complex operation, and it is not directly relevant to the problem at hand, but it is worth noting that it involves dividing one polynomial by another, similar to long division with numbers. Understanding these fundamental concepts of polynomials and their operations is crucial for tackling more complex problems, such as the one we will address in this article.

Problem Statement: Simplifying AB - C

Now that we have a solid foundation in polynomials and their operations, let's address the specific problem at hand. We are given three polynomials:

$A = n$
$B = 2n + 6$
$C = n^2 - 1$

The task is to find the simplest form of the expression $AB - C$ . This involves performing two operations: first, we need to multiply polynomials $A$ and $B$ , and then we need to subtract polynomial $C$ from the result. To solve this problem, we will follow a step-by-step approach. First, we will substitute the given expressions for $A$ , $B$ , and $C$ into the expression $AB - C$ . This will give us an expression involving only the variable $n$ . Next, we will perform the multiplication of $A$ and $B$ , using the distributive property as needed. This will result in a new polynomial expression. Finally, we will subtract $C$ from the result of the multiplication, combining like terms to simplify the expression. The final result will be the simplest form of the expression $AB - C$ , which is a polynomial in $n$ . This problem highlights the importance of understanding polynomial multiplication and subtraction, as well as the ability to simplify algebraic expressions by combining like terms. By carefully following these steps, we can arrive at the correct solution and gain a deeper understanding of polynomial operations.

Step-by-Step Solution

To find the simplest form of $AB - C$ , we will proceed step by step, carefully applying the rules of polynomial operations. This methodical approach ensures accuracy and clarity in our solution. Let's break down the process:

Step 1: Substitute the given polynomials

The first step is to substitute the expressions for $A$ , $B$ , and $C$ into the expression $AB - C$ . We are given:

$A = n$
$B = 2n + 6$
$C = n^2 - 1$

Substituting these values, we get:

AB - C = (n)(2n + 6) - (n^2 - 1)

This substitution transforms the original expression into a concrete form that we can now work with. It sets the stage for the next steps, where we will perform the multiplication and subtraction operations.

Step 2: Multiply A and B

Next, we need to multiply the polynomials $A$ and $B$ . In this case, $A = n$ and $B = 2n + 6$ . To multiply these, we use the distributive property, which states that $a(b + c) = ab + ac$ . Applying this property, we get:

n(2n + 6) = n(2n) + n(6) = 2n^2 + 6n

This multiplication step is crucial as it combines the two polynomials $A$ and $B$ into a single polynomial expression. The result, $2n^2 + 6n$ , is a quadratic polynomial in $n$ .

Step 3: Substitute the result back into the expression

Now that we have the product of $A$ and $B$ , we substitute this result back into the expression $AB - C$ . We have:

AB - C = (2n^2 + 6n) - (n^2 - 1)

This substitution brings us closer to the final simplified form. We now have a polynomial expression that involves subtraction, which we will address in the next step.

Step 4: Subtract C from AB

The final step is to subtract the polynomial $C$ from the product $AB$ . We have:

(2n^2 + 6n) - (n^2 - 1)

To subtract polynomials, we distribute the negative sign to each term in the polynomial being subtracted and then combine like terms. This gives us:

2n^2 + 6n - n^2 + 1

Now, we combine the like terms:

(2n^2 - n^2) + 6n + 1 = n^2 + 6n + 1

This final simplification step yields the simplest form of the expression $AB - C$ , which is a quadratic polynomial in $n$ .

Final Answer

After performing the steps of substitution, multiplication, and subtraction, we have arrived at the simplest form of the expression $AB - C$ . The final answer is:

n^2 + 6n + 1

This is a quadratic polynomial in $n$ , and it represents the result of the original expression $AB - C$ in its most simplified form. This solution demonstrates the importance of following the correct order of operations and carefully combining like terms to simplify polynomial expressions. The process we followed can be applied to a wide range of similar problems involving polynomial operations. Understanding these operations is crucial for success in algebra and related fields.

Key Concepts and Takeaways

Throughout this article, we have explored the process of simplifying polynomial expressions, specifically focusing on the operation $AB - C$ . Let's recap the key concepts and takeaways from this discussion. Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. They are fundamental in algebra and are used extensively in various mathematical and scientific fields.

Polynomial operations, such as addition, subtraction, and multiplication, are essential skills in algebra. When adding or subtracting polynomials, we combine like terms, which are terms with the same variable raised to the same power. When multiplying polynomials, we use the distributive property to multiply each term in one polynomial by each term in the other polynomial.

The problem we addressed involved simplifying the expression $AB - C$ , where $A$ , $B$ , and $C$ were given polynomials. To solve this problem, we followed a step-by-step approach:

Substitute the given expressions for $A$ , $B$ , and $C$ into the expression $AB - C$ .
Multiply polynomials $A$ and $B$ , using the distributive property as needed.
Substitute the result of the multiplication back into the expression.
Subtract $C$ from the result of the multiplication, combining like terms to simplify the expression.

By carefully following these steps, we arrived at the simplest form of the expression $AB - C$ , which was $n^2 + 6n + 1$ . This solution highlights the importance of understanding polynomial multiplication and subtraction, as well as the ability to simplify algebraic expressions by combining like terms. It also emphasizes the need to follow the correct order of operations to ensure accuracy.

The key takeaway from this article is that simplifying polynomial expressions involves a systematic approach that includes substitution, multiplication, and combining like terms. By mastering these techniques, you can confidently tackle a wide range of problems involving polynomial operations. Understanding polynomials and their operations is crucial for success in algebra and related fields, and it forms the foundation for more advanced mathematical concepts.

Practice Problems

To solidify your understanding of simplifying polynomial expressions, it's essential to practice applying the concepts and techniques we've discussed. Here are a few practice problems that are very similar to the one that we just went through that you can tackle on your own. These problems will give you the opportunity to work through the steps of substitution, multiplication, and combining like terms, and they will help you build your confidence in performing polynomial operations.

Problem 1:

Given the polynomials:

$A = x$
$B = 3x - 2$
$C = x^2 + 5$

Find the simplest form of the expression $AB - C$ .

Problem 2:

Given the polynomials:

$P = 2y$
$Q = y + 4$
$R = y^2 - 3y + 1$

Find the simplest form of the expression $PQ - R$ .

Problem 3:

Given the polynomials:

$M = z - 1$
$N = z + 2$
$O = z^2 - 4$

Find the simplest form of the expression $MN - O$ .

For each problem, follow the steps we outlined earlier: substitute the given polynomials into the expression, perform the multiplication, and then subtract the third polynomial, combining like terms to simplify the result. Working through these practice problems will help you develop your skills in polynomial operations and prepare you for more complex algebraic challenges. Be sure to check your answers carefully, and if you encounter any difficulties, review the concepts and examples discussed in this article. Practice is key to mastering polynomial operations and building a strong foundation in algebra.