Simplifying And Classifying Polynomials 4x(x+1)-(3x-8)(x+4)

Simplifying algebraic expressions and classifying polynomials are fundamental concepts in mathematics. This article delves into the step-by-step process of simplifying the expression $4x(x+1)-(3x-8)(x+4)$ and subsequently classifying the resulting polynomial. We will explore the techniques of expanding expressions, combining like terms, and identifying the degree and number of terms in a polynomial. This comprehensive guide aims to provide a clear understanding of these concepts, enhancing your algebraic skills and problem-solving abilities.

Simplifying the Expression $4x(x+1)-(3x-8)(x+4)$

To effectively simplify the expression, we will systematically apply the distributive property and combine like terms. The initial expression is $4x(x+1)-(3x-8)(x+4)$. This involves expanding the products and then reducing the expression to its simplest form. This section will walk you through each step to ensure a clear understanding of the process.

Step 1: Expand the First Term

The first term in the expression is $4x(x+1)$. To expand this, we apply the distributive property, which states that $a(b+c) = ab + ac$. In this case, we multiply $4x$ by both $x$ and $1$:

$4x(x+1) = 4x * x + 4x * 1 = 4x^2 + 4x$

This step transforms the first part of the expression into a more manageable form, setting the stage for further simplification.

Step 2: Expand the Second Term

The second term is $-(3x-8)(x+4)$. Expanding this requires multiplying two binomials. We can use the FOIL method (First, Outer, Inner, Last) to ensure each term in the first binomial is multiplied by each term in the second binomial:

• First: $(3x) * (x) = 3x^2$
• Outer: $(3x) * (4) = 12x$
• Inner: $(-8) * (x) = -8x$
• Last: $(-8) * (4) = -32$

Combining these, we get:

$(3x-8)(x+4) = 3x^2 + 12x - 8x - 32$

Simplifying further by combining like terms ($12x$ and $-8x$), we have:

$3x^2 + 12x - 8x - 32 = 3x^2 + 4x - 32$

Remember, the original expression includes a negative sign in front of this term, so we need to distribute the negative sign:

$-(3x^2 + 4x - 32) = -3x^2 - 4x + 32$

This step is crucial as it correctly expands the second part of the expression while accounting for the negative sign, which is essential for the subsequent steps.

Step 3: Combine the Expanded Terms

Now that we have expanded both parts of the expression, we can combine them:

$4x^2 + 4x - 3x^2 - 4x + 32$

To simplify this, we combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have $4x^2$ and $-3x^2$, as well as $4x$ and $-4x$:

$(4x^2 - 3x^2) + (4x - 4x) + 32$

Step 4: Simplify by Combining Like Terms

Combining the like terms, we get:

$4x^2 - 3x^2 = x^2$

$4x - 4x = 0$

So, the simplified expression is:

$x^2 + 0 + 32 = x^2 + 32$

Thus, the simplified form of the original expression $4x(x+1)-(3x-8)(x+4)$ is $x^2 + 32$. This simplification makes it easier to classify the polynomial, which is the next step in our analysis. The process of expanding and combining like terms is a cornerstone of algebraic manipulation, and mastering it is crucial for solving more complex problems.

Classifying the Resulting Polynomial

After simplifying the expression to $x^2 + 32$, we now need to classify the resulting polynomial. Polynomials are classified based on two main characteristics: their degree and the number of terms they contain. Understanding these classifications helps in further analysis and manipulation of algebraic expressions. In this section, we will define these characteristics and apply them to our simplified polynomial.

Understanding the Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial. In other words, it's the exponent of the term with the highest power of the variable. The degree provides important information about the behavior and characteristics of the polynomial. For instance, a polynomial of degree 2 is known as a quadratic polynomial, while a polynomial of degree 1 is linear, and a polynomial of degree 3 is cubic.

In our simplified polynomial, $x^2 + 32$, the term with the highest power of the variable is $x^2$. The exponent of $x$ in this term is 2. Therefore, the degree of the polynomial $x^2 + 32$ is 2. This means that the polynomial is a quadratic polynomial.

Understanding the Number of Terms in a Polynomial

The number of terms in a polynomial refers to the individual expressions that are separated by addition or subtraction signs. Each term consists of a coefficient (a numerical value) multiplied by a variable raised to a power, or it may simply be a constant. Classifying polynomials by the number of terms helps in categorizing them into specific types, such as monomials, binomials, and trinomials.

In the polynomial $x^2 + 32$, we have two terms: $x^2$ and $32$. These terms are separated by an addition sign. Therefore, the polynomial has two terms. A polynomial with two terms is called a binomial. This classification is essential for identifying the structure and properties of the polynomial.

Classifying $x^2 + 32$

Now that we have determined the degree and the number of terms, we can classify the polynomial $x^2 + 32$. As we established, the degree of the polynomial is 2, making it a quadratic polynomial. Additionally, it has two terms, which means it is a binomial.

Therefore, the polynomial $x^2 + 32$ is classified as a quadratic binomial. This classification allows us to accurately describe the polynomial based on its mathematical characteristics. The degree and number of terms are fundamental properties that help in understanding the behavior and potential applications of the polynomial.

Identifying the Correct Option

Having simplified the expression $4x(x+1)-(3x-8)(x+4)$ to $x^2 + 32$ and classified it as a quadratic binomial, we can now identify the correct option from the given choices. The options were:

A. Quadratic binomial B. Linear binomial C. Quadratic monomial D. Quadratic trinomial

Analyzing the Options

• Option A: Quadratic binomial – This option correctly identifies the polynomial as having a degree of 2 (quadratic) and two terms (binomial). This matches our classification of $x^2 + 32$.
• Option B: Linear binomial – A linear polynomial has a degree of 1, which does not match our polynomial’s degree of 2. Therefore, this option is incorrect.
• Option C: Quadratic monomial – A monomial has only one term, while our polynomial has two terms ($x^2$ and $32$). Thus, this option is incorrect.
• Option D: Quadratic trinomial – A trinomial has three terms, but our polynomial has only two terms. This option is also incorrect.

Conclusion

Based on our analysis, the correct classification of the simplified polynomial $x^2 + 32$ is a quadratic binomial. Therefore, the correct option is A. Quadratic binomial. This conclusion reaffirms the importance of accurately simplifying expressions and correctly classifying polynomials based on their degree and the number of terms they contain.

Conclusion

In this article, we have successfully simplified the expression $4x(x+1)-(3x-8)(x+4)$ to $x^2 + 32$ and classified the resulting polynomial as a quadratic binomial. This process involved expanding the expression using the distributive property and the FOIL method, combining like terms, and then identifying the degree and number of terms in the simplified polynomial.

We began by expanding the expression step-by-step, ensuring each term was correctly multiplied and accounted for. This meticulous approach is crucial in algebraic manipulations to avoid errors and maintain accuracy. The expansion process demonstrated the practical application of algebraic principles, such as the distributive property, which is a cornerstone of algebraic simplification.

Next, we combined like terms, which involved identifying terms with the same variable raised to the same power and adding or subtracting their coefficients. This step is vital for reducing the expression to its simplest form, making it easier to analyze and classify. The simplified form, $x^2 + 32$, clearly shows the polynomial's structure and properties.

Classification of the polynomial followed, where we defined the degree and the number of terms. The degree, being the highest power of the variable, was identified as 2, indicating a quadratic polynomial. The number of terms, which was two, classified it as a binomial. Combining these two characteristics led to the final classification of the polynomial as a quadratic binomial.

Finally, we matched our classification with the given options, confidently selecting option A, the quadratic binomial. This final step underscored the importance of accurate classification in mathematical problem-solving. The entire process, from simplification to classification, reinforces fundamental algebraic skills that are essential for more advanced mathematical studies.

By understanding how to simplify expressions and classify polynomials, you are better equipped to tackle more complex algebraic problems. This knowledge not only enhances your mathematical skills but also improves your problem-solving abilities in various other fields. The ability to manipulate and classify algebraic expressions is a valuable asset in mathematics and beyond.