Simplify And Classify Polynomial Expression 4x(x+1)-(3x-8)(x+4)

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In this detailed guide, we will walk through the process of simplifying the polynomial expression 4x(x+1)-(3x-8)(x+4) and then classify the resulting polynomial. This involves expanding the expression, combining like terms, and identifying the degree and number of terms to determine the polynomial's classification. Polynomials are fundamental in algebra, and understanding how to simplify and classify them is crucial for various mathematical applications. Let's dive into the step-by-step solution.

Step 1: Expand the Expression

The first step in simplifying the expression 4x(x+1)-(3x-8)(x+4) is to expand the terms. This involves applying the distributive property and the FOIL (First, Outer, Inner, Last) method. We begin by expanding the first term, 4x(x+1):

  • Distribute 4x across (x+1): 4x * x + 4x * 1 = 4x^2 + 4x

Next, we expand the second term, (3x-8)(x+4). This requires using the FOIL method, which ensures that each term in the first binomial is multiplied by each term in the second binomial:

  • FOIL Method:
    • First: 3x * x = 3x^2
    • Outer: 3x * 4 = 12x
    • Inner: -8 * x = -8x
    • Last: -8 * 4 = -32

Combining these results, we get:

3x^2 + 12x - 8x - 32

Now, simplify by combining like terms (12x and -8x):

3x^2 + 4x - 32

So, the expanded form of (3x-8)(x+4) is 3x^2 + 4x - 32.

Step 2: Combine the Expanded Terms

Now that we have expanded both parts of the original expression, we can combine them. The original expression is 4x(x+1)-(3x-8)(x+4), and we have expanded it to:

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

To combine these terms, we need to distribute the negative sign across the second expression:

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

Now, group like terms together:

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

Step 3: Simplify by Combining Like Terms

Combine the like terms to simplify the expression:

  • Combine x^2 terms: 4x^2 - 3x^2 = x^2
  • Combine x terms: 4x - 4x = 0
  • The constant term: 32

So, the simplified expression is:

x^2 + 0 + 32

Which simplifies further to:

x^2 + 32

Step 4: Classify the Resulting Polynomial

Now that we have simplified the expression to x^2 + 32, we need to classify it. Polynomials are classified based on their degree and the number of terms they contain.

Degree of the Polynomial

The degree of a polynomial is the highest power of the variable in the expression. In this case, the highest power of x is 2 (x^2), so the degree of the polynomial is 2. A polynomial of degree 2 is called a quadratic polynomial.

Number of Terms

The number of terms in a polynomial is the number of individual expressions separated by addition or subtraction. In our simplified expression, x^2 + 32, there are two terms: x^2 and 32. A polynomial with two terms is called a binomial.

Classification

Based on the degree and the number of terms, we can classify the polynomial x^2 + 32 as a quadratic binomial.

Detailed Explanation of Polynomial Classification

To further understand the classification, let's delve deeper into the definitions and examples of different types of polynomials.

Polynomial Basics

A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Polynomials are a fundamental concept in algebra and are used extensively in various mathematical fields.

A general form of a polynomial can be written as:

a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0

Where:

  • x is the variable.
  • n is a non-negative integer representing the degree of the term.
  • a_n, a_{n-1}, ..., a_1, a_0 are the coefficients, which are constants.

Classification by Degree

The degree of a polynomial is the highest power of the variable in the polynomial. Polynomials are classified by their degree as follows:

  • Constant Polynomial: A polynomial with degree 0. It is a constant value (e.g., 5, -3, 1/2).
  • Linear Polynomial: A polynomial with degree 1 (e.g., 2x + 1, -x + 4).
  • Quadratic Polynomial: A polynomial with degree 2 (e.g., x^2 + 3x - 2, 4x^2 - 7).
  • Cubic Polynomial: A polynomial with degree 3 (e.g., x^3 - 2x^2 + x - 5, 2x^3 + 9).
  • Quartic Polynomial: A polynomial with degree 4 (e.g., x^4 + 2x^3 - x^2 + 3x + 1).
  • Quintic Polynomial: A polynomial with degree 5 (e.g., x^5 - 3x^4 + 2x^3 - x^2 + 4x - 6).

Classification by Number of Terms

The number of terms in a polynomial is the count of individual expressions separated by addition or subtraction. Polynomials are classified by the number of terms as follows:

  • Monomial: A polynomial with one term (e.g., 3x^2, -5x, 7).
  • Binomial: A polynomial with two terms (e.g., x^2 + 32, 2x - 5, x^3 + 1).
  • Trinomial: A polynomial with three terms (e.g., x^2 + 3x - 2, 4x^2 - x + 7, x^3 + 2x^2 - 1).
  • Polynomials with four or more terms are generally referred to as polynomials with that number of terms (e.g., a polynomial with four terms).

Examples of Polynomial Classification

Let's look at some examples to illustrate the classification of polynomials:

  1. 5x^3 - 2x + 1: This is a cubic trinomial because it has a degree of 3 and three terms.
  2. 4x^2 - 9: This is a quadratic binomial because it has a degree of 2 and two terms.
  3. 7x: This is a linear monomial because it has a degree of 1 and one term.
  4. 3: This is a constant monomial because it has a degree of 0 and one term.
  5. x^4 - 2x^3 + x^2 - 5x + 4: This is a quartic polynomial with five terms.

Why is Polynomial Simplification and Classification Important?

Polynomials are used extensively in mathematics, science, engineering, and economics. Simplifying and classifying polynomials is important for several reasons:

  • Solving Equations: Simplifying polynomials is often a necessary step in solving polynomial equations. By reducing a complex polynomial to its simplest form, it becomes easier to find the roots or solutions of the equation.
  • Graphing Functions: The degree and coefficients of a polynomial provide valuable information about the shape and behavior of the polynomial function's graph. Knowing the classification helps in sketching the graph and understanding its key features.
  • Calculus: Polynomials are the building blocks of many functions in calculus. Understanding polynomial operations is essential for differentiation and integration.
  • Modeling Real-World Phenomena: Polynomials are used to model various real-world phenomena, such as projectile motion, economic growth, and population dynamics. Simplifying these models helps in making predictions and understanding the underlying processes.

Conclusion

In conclusion, simplifying the expression 4x(x+1)-(3x-8)(x+4) leads to the polynomial x^2 + 32. This polynomial is classified as a quadratic binomial because it has a degree of 2 and two terms. The process involved expanding the expression, combining like terms, and then identifying the degree and number of terms. Understanding polynomial simplification and classification is a fundamental skill in algebra and has wide-ranging applications in mathematics and other fields. By mastering these concepts, you can tackle more complex mathematical problems and gain a deeper understanding of algebraic structures. The ability to simplify and classify polynomials equips you with essential tools for problem-solving in various contexts, reinforcing the importance of these foundational algebraic skills.