Simplifying Polynomial Expressions A Comprehensive Guide To AB + C

Let's embark on a journey into the realm of polynomials, where we encounter three distinct entities: A, B, and C. These polynomials, each a unique expression composed of variables and coefficients, hold the key to unlocking a fascinating mathematical puzzle. Our primary focus is to unravel the enigma of $AB + C$, seeking its simplest form. To accomplish this, we must first meticulously define each polynomial, paving the way for our algebraic exploration.

A, the first polynomial in our equation, is represented by the expression $x + 1$. This linear expression, characterized by its variable x raised to the power of 1, forms the foundation of our polynomial trio. The coefficient of x is 1, and the constant term is also 1. This simplicity belies its significance, as it will play a crucial role in the subsequent calculations.

Next, we encounter B, a polynomial of higher degree. Represented by the expression $x^2 + 2x - 1$, B is a quadratic polynomial, owing to the presence of the $x^2$ term. This quadratic expression introduces an element of complexity, requiring careful consideration of its coefficients and terms. The coefficient of $x^2$ is 1, the coefficient of x is 2, and the constant term is -1. Understanding the interplay of these terms is paramount to simplifying the expression $AB + C$.

Finally, we arrive at C, a monomial consisting of a single term: $2x$. This linear expression, similar to A, possesses a variable x raised to the power of 1. However, C lacks a constant term, distinguishing it from A. The coefficient of x in C is 2, further emphasizing its unique identity within our polynomial ensemble.

With the individual polynomials A, B, and C clearly defined, we are now poised to delve into the heart of the matter: determining the simplest form of $AB + C$. This requires a systematic approach, employing the principles of polynomial multiplication and addition. By meticulously applying these algebraic techniques, we shall unravel the complexities of the expression and arrive at its most concise representation.

Step-by-Step Multiplication of Polynomials A and B

To evaluate the expression $AB + C$, our first crucial step involves multiplying the polynomials A and B. This process, often referred to as polynomial multiplication, requires careful application of the distributive property. The distributive property, a cornerstone of algebra, dictates that each term in the first polynomial must be multiplied by each term in the second polynomial. This systematic approach ensures that no term is overlooked, leading to an accurate product.

Let's embark on this multiplication journey, where A is represented by $(x + 1)$ and B by $(x^2 + 2x - 1)$. We begin by distributing the first term of A, which is x, across all terms of B. This yields the following partial products:

• $x * x^2 = x^3$
• $x * 2x = 2x^2$
• $x * -1 = -x$

Next, we distribute the second term of A, which is 1, across all terms of B. This results in the following partial products:

• $1 * x^2 = x^2$
• $1 * 2x = 2x$
• $1 * -1 = -1$

Now, we have a collection of partial products. To obtain the complete product of A and B, we must combine these partial products. This involves adding the terms with the same degree of x. In other words, we group together the cubic terms ($x^3$), the quadratic terms ($x^2$), the linear terms (x), and the constant terms. This systematic grouping simplifies the process of combining like terms.

Combining the partial products, we get:

$x^3 + 2x^2 - x + x^2 + 2x - 1$

Now, we identify and combine like terms. We have one cubic term ($x^3$), which remains unchanged. We have two quadratic terms ($2x^2$ and $x^2$), which combine to give $3x^2$. We have two linear terms (-x and 2x), which combine to give x. Finally, we have one constant term (-1), which also remains unchanged.

Therefore, the product of A and B, denoted as AB, is:

$x^3 + 3x^2 + x - 1$

This expression represents the first crucial component in our quest to simplify $AB + C$. With AB now determined, we are well-prepared to proceed to the next step: adding polynomial C to this product.

Adding Polynomial C to the Product AB

Having successfully computed the product of polynomials A and B, our attention now shifts to the final piece of the puzzle: adding polynomial C to the product AB. This process, known as polynomial addition, involves combining like terms from both expressions. Like terms, as the name suggests, are terms that share the same variable and exponent. For instance, $3x^2$ and $-5x^2$ are like terms, while $3x^2$ and $3x$ are not.

Recall that we previously determined the product AB to be $x^3 + 3x^2 + x - 1$. We also know that polynomial C is given by $2x$. To add C to AB, we simply write the two expressions side-by-side and combine the like terms.

$(x^3 + 3x^2 + x - 1) + (2x)$

Now, we identify the like terms. We have one cubic term ($x^3$), one quadratic term ($3x^2$), two linear terms (x and $2x$), and one constant term (-1). To combine the like terms, we add their coefficients. The coefficient of a term is the numerical factor that multiplies the variable part.

• The cubic term $x^3$ has a coefficient of 1 and remains unchanged.
• The quadratic term $3x^2$ has a coefficient of 3 and also remains unchanged.
• The linear terms x and $2x$ have coefficients of 1 and 2, respectively. Adding these coefficients gives us $1 + 2 = 3$, so the combined linear term is $3x$.
• The constant term -1 remains unchanged.

Combining these results, we obtain the final expression for $AB + C$:

$x^3 + 3x^2 + 3x - 1$

This expression represents the simplest form of $AB + C$. It is a cubic polynomial, characterized by its highest degree term, $x^3$. The polynomial is written in standard form, with the terms arranged in descending order of their exponents. This meticulous arrangement enhances clarity and facilitates further algebraic manipulations.

Solution and Final Answer

After a meticulous journey through polynomial multiplication and addition, we have successfully simplified the expression $AB + C$. Our step-by-step approach, grounded in the fundamental principles of algebra, has led us to a clear and concise solution. Let's recap our findings and present the final answer.

We began by defining the polynomials A, B, and C as follows:

• $A = x + 1$
• $B = x^2 + 2x - 1$
• $C = 2x$

Our primary objective was to determine the simplest form of the expression $AB + C$. To achieve this, we first multiplied polynomials A and B, employing the distributive property to ensure accuracy. This yielded the product AB:

$AB = x^3 + 3x^2 + x - 1$

Next, we added polynomial C to the product AB. This involved combining like terms, a process that required careful attention to coefficients and exponents. The result of this addition was the simplified expression for $AB + C$:

$AB + C = x^3 + 3x^2 + 3x - 1$

This cubic polynomial, $x^3 + 3x^2 + 3x - 1$, represents the simplest form of the expression $AB + C$. It is written in standard form, with the terms arranged in descending order of their exponents. This arrangement enhances clarity and facilitates further algebraic manipulations.

Therefore, the final answer to our problem is:

$x^3 + 3x^2 + 3x - 1$

This solution aligns perfectly with option C in the original problem statement. Our meticulous step-by-step approach has not only yielded the correct answer but also provided a comprehensive understanding of the underlying algebraic principles. This mastery of polynomial manipulation will undoubtedly serve as a valuable asset in future mathematical endeavors.

In conclusion, we have successfully navigated the complexities of polynomial algebra, unraveling the expression $AB + C$ and arriving at its simplest form. Our journey, marked by careful application of algebraic principles, has culminated in a clear and concise solution. The expression $x^3 + 3x^2 + 3x - 1$ stands as a testament to the power of systematic problem-solving and the beauty of mathematical precision.

In summary, our exploration of the polynomial expression $AB + C$ has been a rewarding journey through the realm of algebra. We have meticulously dissected the problem, applying the principles of polynomial multiplication and addition to arrive at the simplified form: $x^3 + 3x^2 + 3x - 1$. This solution, corresponding to option C, stands as a testament to the power of systematic problem-solving and the elegance of mathematical precision. The skills and insights gained during this exploration will undoubtedly prove invaluable in tackling future algebraic challenges. Understanding polynomial operations is crucial, and this step-by-step guide provides a solid foundation. This mathematical simplification showcases the beauty of algebraic manipulation.