One way to determine the factors of x3 – 12×2 – 2x + 24 by grouping is as follows:

1. Group the first two terms (x3 – 12×2) and the last two terms (-2x + 24) together.

2. Factor out the greatest common factor from each group. In this case, x2 is the greatest common factor of the first group, and -2 is the greatest common factor of the second group.

3. Rewrite the polynomial using the factored form of each group. (x2(x – 12) – 2(x – 12))

4. Factor out the common binomial factor (x – 12).

5. The final factored form is (x – 12)(x2 – 2).

Factoring polynomials not only helps you simplify complex expressions but also helps you solve polynomial equations efficiently.

## Understanding the Basics of Factoring Polynomials

Factoring polynomials involves determining the factors of an equation. Understanding the basics of factoring polynomials is important for problem-solving and algebraic manipulations. In this article, we will discuss the process of factoring polynomials and take a look at one way to determine the factors of the equation x3 – 12×2 – 2x + 24 by grouping.

### Reviewing polynomial terminology

Polynomial terminology can be confusing, but a basic understanding of the terms makes factoring polynomials much easier.

Some commonly used terms in polynomial expressions are:

Term Definition

Terms: Expressions separated by operators like plus or minus

Coefficient: The numerical factor of the term, written before the variable

Degree: The highest exponent in the term or expression

Monomial: A term with only one variable

Binomial: A term with two variables or two terms connected by an operator

Trinomial: A term with three variables or three terms connected by operators

By understanding these terms, factoring polynomials like x3 – 12×2 – 2x + 24 becomes a matter of identifying the common factors and grouping them accordingly.

Factoring polynomials might seem intimidating initially, but with a little practice it becomes easy to get the hang of.

### Understanding the purpose of factoring polynomials

Factoring polynomials is a fundamental concept in algebra that involves breaking down a polynomial expression into simpler components. The purpose of factoring polynomials is to simplify expressions, solve algebraic equations, and identify the roots of a polynomial function.

Let’s take the example of the polynomial expression “x3 – 12×2 – 2x + 24”. Factoring this expression can help us find its roots and solve algebraic equations involving it. One way to factor this expression is by grouping, which involves rearranging terms in a polynomial and then factoring out the common factor from each set of terms. The result is a simplified expression that is easier to work with and can be used to solve various algebraic problems.

Pro tip: Practice factoring polynomials with different methods to better understand each technique and how it can be applied to different expressions.

### Introduction to factoring by grouping

Factoring by grouping is a technique used in algebra to find the factors of a polynomial. It involves grouping terms with common factors and factoring out these common factors. To understand factoring by grouping, let’s take the example of x³ – 12x² – 2x + 24.

Group the first two terms (x³ – 12x²) and factor out the common factor (x²). This will give you x²(x – 12).

Group the last two terms (-2x + 24) and factor out the common factor (-2). This will give you -2(x – 12).

Notice that both groups have (x – 12) as a common factor. Combine the two groups and factor out (x – 12) to get (x² – 2)(x – 12).

Factoring by grouping may seem daunting initially, but it can become an essential tool in solving more complex algebraic equations with practice.

## Identifying the Polynomial and Its Degree

A polynomial is a sum of variables raised to the power of non-negative integers in mathematics. For example, in the polynomial x3 – 12×2 – 2x + 24, the variables are ‘x’ and the polynomial terms can be identifed as x3, -12×2, -2x and 24. It also has a degree of 3, as the highest power of the variable is 3. The aim in factoring polynomials is to identify its factors. Let’s look at one way to determine the factors of x3 – 12×2 – 2x + 24 by grouping.

### Understanding degrees of polynomials

In algebra, a polynomial is an expression consisting of variables and coefficients, and having one or more terms. The degree of a polynomial is the highest power (exponent) of the variable in the expression.

For example, the polynomial expression 3×2 + 2x – 1 has a degree of 2 because the highest exponent of x is 2.

To determine the degree of a polynomial, identify the term with the highest exponent of the variable in the expression.

In the polynomial x3 – 2×2 + 5x – 7, the term with the highest exponent is x3, so the polynomial degree is 3.

Understanding the degree of a polynomial is important in algebra because it can help you determine the complexity of the solution and the number of solutions that exist.

Factoring polynomials is one way to determine the factors of a polynomial expression, such as x3 – 12×2 – 2x + 24. Grouping similar terms can simplify the expression and make it easier to factor. Pro Tip: Practice identifying the degree of polynomials until you are comfortable with the concept.

### Identifying the degree of x3 – 12×2 – 2x + 24

The given expression, x3 – 12×2 – 2x + 24, is a polynomial with four terms and a degree of 3.

To identify the degree of a polynomial, we need to find the term with the highest power of the variable. In this case, the highest power of x is 3, which means that the degree of the polynomial is 3.

To factor this polynomial using grouping, we can first group the first two terms and the last two terms:

(x3 – 12×2) + (-2x + 24)

Next, we can factor out the greatest common factor from each group:

x2(x – 12) – 2(x – 12)

Now, we can see that we have a common binomial factor of (x – 12), which we can factor out:

(x2 – 2)(x – 12)

Therefore, the factors of x3 – 12×2 – 2x + 24 are (x2 – 2) and (x – 12).

Pro tip: Identifying the degree of a polynomial is crucial in determining its end behavior, which can help us understand the shape of its graph. Factoring polynomials can also help solve equations and simplify algebraic expressions.

### Exploring common factors

To factorize a polynomial, the first step is to determine the polynomial’s degree and identify the common factor(s) that can be removed. This technique can be helpful to factorize more complex polynomials.

Let’s take the example of x3 – 12×2 – 2x + 24, to determine its factors by grouping. We will take the common factor from the first two terms (x2) and the last two (-2). By doing so, we get:

x2(x – 12) – 2(x -12)

You can see that both terms have a common factor – (x – 12). So, we can write:

(x – 12)(x2 – 2)

Therefore, the factors of x3 – 12×2 – 2x + 24 are (x – 12) and (x2 – 2).

We can factorize more complex polynomials with higher degrees and multiple terms using this technique.

Pro Tip: Practice factoring simpler polynomials before moving on to more complex ones. It will help you to understand and master the concept of factoring.

## Factoring by Grouping

Factoring by grouping is one of the most effective methods for factoring polynomials and can be used to determine the factors of x3 – 12×2 – 2x + 24. This method involves identifying, classifying, and grouping the terms in the polynomial to create smaller polynomials. This section will discuss the steps and processes necessary to factor polynomials using this method.

### Identifying the factors and grouping

Factoring by grouping is a method used to factor polynomials with four or more terms by grouping the terms into pairs and factoring out a common factor. Here’s how to factor x³ – 12x² – 2x + 24 by grouping:

1. Group the first two terms and the last two terms:

(x³ – 12x²) – (2x – 24)

2. Factor out the common factors from each group:

x²(x – 12) – 2(x – 12)

3. Note that the remaining terms have a common factor of (x – 12):

(x² – 2)(x – 12)

4. Thus, x³ – 12x² – 2x + 24 can be factored as (x² – 2)(x – 12) through factoring by grouping.

Factoring by grouping may initially seem complicated, but it is a useful method for simplifying complex polynomials.

### Simplifying the grouping

Factoring by grouping is a technique used to factor polynomials that involve four terms. This technique simplifies the grouping by grouping the polynomials into two groups of two terms. Here are the steps to factor the polynomial x3 – 12×2 – 2x + 24 by grouping:

Step Process

1 Group the first two terms and factor out the greatest common factor. x²(x – 12).

2 Group the last two terms and factor out the greatest common factor. -2(x – 12).

3 Combine the two factored groups. x²(x – 12) – 2(x – 12).

4 Factor out the common factor (x – 12) from the expression. (x – 12)(x² – 2).

By simplifying the grouping, we can easily determine the given polynomial factors, which are (x – 12) and (x² – 2).

Pro tip: Practice makes perfect when it comes to factoring by grouping. The more polynomials you factor, the easier it becomes to identify the right grouping.

### Finalizing the factors for x3 – 12×2 – 2x + 24

To factorize a polynomial expression like x3 – 12×2 – 2x + 24, we can use the technique of Factoring by Grouping. This involves grouping the terms in the polynomial in pairs and factoring out each pair’s greatest common factor, allowing us to factor out the remaining factors in the expression.

Here’s how to factorize x3 – 12×2 – 2x + 24 using the method of Factoring by Grouping:

Step 1: Group the first two terms and the last two terms together, like so: (x3 – 12×2) + (-2x + 24)

Step 2: Factor out the GCF of each pair: x2(x – 12) – 2(x – 12)

Step 3: Factor out the common binomial factor of (x – 12): (x – 12)(x2 – 2)

Therefore, the factors of the polynomial x3 – 12×2 – 2x + 24 are (x – 12)(x2 – 2). By factoring the expression, we can simplify it and find the equation’s roots, or solutions, more easily.

Pro Tip: Factoring by Grouping is a helpful technique for factorizing polynomial expressions that have four terms. Remember to group the terms carefully and look for the GCF of each pair before factoring out the common binomial factor.

## Which Shows One Way to Determine the Factors of x3 – 12×2 – 2x + 24 By Grouping?

Factoring polynomials can be done in several different ways. One of these ways is by determining the factors of x3 – 12×2 – 2x + 24 by grouping. In addition, this factoring method can be used to solve for the zeros of the polynomial, which can be found using the quadratic equation.

In this article, we will explain how to solve for the zeros of the polynomial by grouping and other methods.

### Completing the factored form

Completing the factored form is often necessary to solve for the zeros of a polynomial. One method for determining the factors of a polynomial is grouping, which can be applied to polynomials with four or more terms.

Here’s how to factor a polynomial using grouping:

1. Group the first and second terms of the polynomial terms separately.

2. Factor out the greatest common factor of each grouping.

3. Identify the binomial factors that are common to both groupings.

4. Rewrite the polynomial in factored form using the common binomial factors.

For example, consider the polynomial x^3 − 12x^2 − 2x + 24. Grouping the terms separately, we get (x^3 − 12x^2) + (−2x + 24). Factoring out the greatest common factor of each grouping, we get x^2(x − 12) − 2(x − 12). Identifying the common binomial factor (x − 12), we can rewrite the polynomial in factored form as (x − 12) (x^2 − 2). Pro tip: practicing factoring polynomials using different methods can improve your algebra skills and make solving more complex problems easier.

### Solving for the roots

To find the zeros or roots of a polynomial, you can use the technique of factoring. For example, grouping is one way to determine the factors of a polynomial like x3 – 12×2 – 2x + 24.

Here are the steps to follow to solve for the roots:

Step Description

1 Group the first two terms and the last two terms.

2 Factor out the common terms from each group.

3 Factor out the common binomial factor from the resulting expression.

4 Factor the remaining binomial.

5 Set each factor equal to zero and solve for x to find the roots of the polynomial.

By grouping the polynomial terms, factoring can be simplified and the roots can be found more efficiently.

### Checking the results

Checking the results is an important step in solving for the zeros of the polynomial or factoring polynomials using grouping method. To check the results of finding the zeros, you can use the remainder theorem or synthetic division to plug in the possible roots and see if they result in a remainder of zero.

To check the results of factoring a polynomial, multiply the factors you have determined to see if they result in the original polynomial. It is also helpful to check for common mistakes such as forgetting to distribute a negative sign or to combine like terms.

Checking the results not only ensures that you have found the correct zeros or factors but also allows you to catch any mistakes and make adjustments before submitting your final answer or moving on to the next step of the problem-solving process.

## Real-world Application

Factoring polynomials is a calculation method that has plenty of real-world applications. It can simplify equations, solve problems, and even determine the factors of a certain term.

In this article, we will look at one of the ways to factor polynomials and there will be a real-world application to understand the concept better.

### Applying factoring polynomials in business and finance

Factoring polynomials is a crucial problem-solving skill in business and finance that requires breaking down complex mathematical expressions into simpler components. For example, a common real-world application of factoring involves determining the factors of x3 – 12×2 – 2x + 24 by grouping.

Here is how to perform this process:

First, group the terms with common factors. This gives (x3-2x) – (12×2-24). Then, we factor out the common terms. (x(x2-2)) – (12(x2-2)). Next, notice that (x2-2) is a common factor in both groups, so we can combine terms to get (x-12)(x2-2).

By applying factoring polynomials to real-world problems in business and finance, one can simplify complex expressions, solve equations, and make informed decisions based on the data at hand.

### Real-life examples of factoring by grouping

Factoring by grouping is a useful technique for solving polynomials and is widely used in real-world applications. For example, when determining the factors of x3 – 12×2 – 2x + 24, you can use this method to break down the polynomial into two smaller polynomials.

Here’s how to do it:

Group the first two terms and the last two terms together.

Factor out the Greatest Common Factor (GCF) of each group.

Notice that the two groups share a common factor (x – 4).

Factor out the shared factor.

Simplify the expression.

This technique is used in various fields such as engineering, physics, economics, and computer science. It helps in understanding the behavior of complex systems by breaking them down into simpler components. Pro Tip: Understanding factoring by grouping is essential for solving advanced-level mathematical problems.

### The importance of factoring polynomials in higher mathematics

Factoring polynomials is essential in higher mathematics, with real-world applications in engineering, physics, and economics. For example, polynomials represent relationships between variables in mathematical models, making it easier to analyze and understand complex systems.

One way to factor polynomials is by using a technique called grouping, which involves breaking down the polynomial into pairs of terms with a common factor. For instance, the polynomial x3 – 12×2 – 2x + 24 can be factored into (x2 – 2)(x – 12) by grouping the terms as follows:

(x3 – 2x) – (12×2 – 24)

= x(x2 – 2) – 12(x2 – 2)

= (x – 12)(x2 – 2)

By factoring polynomials, we can solve a wide range of mathematical problems, from finding the roots and zeros of an equation to simplifying complex expressions and identifying patterns in data.