Problem 83 - Elementary Algebra

Question

Factor by grouping. $$ x 3+7 x 2-2 x-14 $$

Step-by-Step Solution

Answer

The factored form is (x + 7)(x^2 - 2).

Step 1: Identify Grouping Pairs

To factor by grouping, we first split the polynomial into two pairs of terms. For this polynomial, we will group x^3 + 7x^2 i and -2x - 14 together.

Step 2: Factor Out the Greatest Common Factor (GCF) from Each Group

In the first group, x^3 + 7x^2, the GCF is x^2, which gives us x^2(x + 7). In the second group, -2x - 14, the GCF is -2, which results in -2(x + 7).

Step 3: Factor Out the Common Binomial

Now observe that both groups contain the common binomial factor (x + 7). We can factor this out to get (x + 7)(x^2 - 2).

Step 4: Write the Final Factored Form

The expression can be written as (x + 7)(x^2 - 2), which is the factorization of the original polynomial by grouping.

Key Concepts

Grouping

Grouping is a technique used to simplify and factor polynomials by collecting terms into smaller, more manageable groups. This method is particularly useful for polynomials with four or more terms.
When you group terms, the idea is to rearrange them in such a way that you can easily extract common factors. By doing so, you essentially create smaller binomials or expressions that have something in common.
In the exercise, we grouped the polynomial terms like this:

• Group the first two terms: \( x^3 + 7x^2 \)
• Group the last two terms: \( -2x - 14 \).

The goal of grouping is to make the polynomial easier to factor by concentrating on these smaller expressions.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor shared by terms in an expression. When factoring by grouping, finding the GCF is essential for simplifying each group of terms.
In our original exercise:

• For the group \(x^3 + 7x^2\), the GCF is \(x^2\), giving us \(x^2(x + 7)\).
• For the group \(-2x - 14\), the GCF is \(-2\), leading to \(-2(x + 7)\).

Finding the GCF in smaller groups allows us to factor the expression further and makes the polynomial simpler to work with. This also provides easier steps for future algebraic manipulation.

Polynomials

A polynomial is an algebraic expression made up of variables and coefficients, involving terms that are only added, subtracted, or multiplied.
Polynomials can have different degrees, which are determined by the highest power of the variable present. The exercise we tackled involves a third-degree polynomial due to the term \(x^3\).
Polynomials can consist of multiple terms, like in our example, which consists of four terms: \(x^3\), \(7x^2\), \(-2x\), and \(-14\).

• These expressions are key to many areas in mathematics.
• They can be manipulated in various ways, such as factoring, expanding, and solving equations.

Understanding polynomials is crucial because they represent numerous real-world situations and can model various types of behavior.

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and operators like addition and multiplication. They form the building blocks of algebra and are used in equations and inequalities.
An algebraic expression can be simple like \(2x\) or complex like in our polynomial example. It's important to

• Recognize different parts of the expression, like terms and coefficients.
• Use operations like addition, subtraction, and multiplication to simplify or factor the expression.

Through understanding and manipulating algebraic expressions, we can solve real-life problems that require mathematical models and calculations, making them invaluable tools in mathematics.