Problem 13 - Elementary Algebra

Question

Factor by grouping. $$ x 3+5 x 2-3 x-15 $$

Step-by-Step Solution

Answer

The factored form is \((x + 5)(x^2 - 3)\).

Step 1: Group the terms

First, rearrange and group the terms of the polynomial into two pairs. The given expression is \(x^3 + 5x^2 - 3x - 15\). You can group them as: \((x^3 + 5x^2) + (-3x - 15)\).

Step 2: Factor out the greatest common factor from each group

In the first group, \(x^3 + 5x^2\), the greatest common factor is \(x^2\). Factoring it out gives \(x^2(x + 5)\). In the second group, \(-3x - 15\), the greatest common factor is \(-3\). Factoring it out gives \(-3(x + 5)\). The expression is now \(x^2(x + 5) - 3(x + 5)\).

Step 3: Factor out the common binomial

Now, observe that both terms \(x^2(x + 5)\) and \(-3(x + 5)\) have a common factor \((x + 5)\). Factor out \((x + 5)\) to get \((x + 5)(x^2 - 3)\).

Step 4: Final check and expression

The expression is now fully factored as \((x + 5)(x^2 - 3)\). Make sure no further factoring can be done, and the expression is simplified.

Key Concepts

Understanding Polynomials

Polynomials are mathematical expressions made up of variables and constants, using operations like addition, subtraction, and multiplication. They are written in terms of powers of a variable. For example, the polynomial in the exercise is expressed as \(x^3 + 5x^2 - 3x - 15\). Here, \(x\) is the variable, and each term, such as \(5x^2\), consists of a coefficient and a variable raised to a power.

• Coefficients: Numbers in front of the variables (e.g., 5 in \(5x^2\)).
• Exponents: Powers to which the variable is raised (e.g., 3 in \(x^3\)).
• Constants: Terms without variables, like -15 in the polynomial.

In algebra, polynomials are often rearranged and factored for simplification or to solve equations. The process of factoring helps to express the polynomial as a product of its simpler components.

Factor by Grouping

Factor by grouping is a method used to factor polynomials, especially when they have four or more terms. This technique involves rearranging the terms and grouping them into pairs, making it easier to identify common factors.
Here are the steps:

• First, organize the polynomial into pairs or groups, as done with \((x^3 + 5x^2) + (-3x - 15)\).
• Then, factor out the greatest common factor from each group. In this exercise, \(x^2\) is taken out from the first group, and \(-3\) from the second group.
• After factoring out, notice any common binomial. In this case, \((x + 5)\) was common.
• Finally, factor that binomial out to achieve the final expression \((x + 5)(x^2 - 3)\).

Factor by grouping is especially useful when dealing with complex polynomials that might not be easily simplified by direct factoring.

Exploring the Greatest Common Factor

The greatest common factor (GCF) is a key concept in algebraic factoring. It refers to the largest factor that is common to all terms in a polynomial. Identifying and factoring out the GCF simplifies expressions and is a crucial step before further factoring methods.
For instance, in the polynomial \(x^3 + 5x^2\), the GCF is \(x^2\), because \(x^2\) is the largest power of \(x\) that divides both terms evenly. Factoring out \(x^2\) gives \(x^2(x + 5)\).

• Identify the GCF: Look for the highest power of variables common in all terms.
• Factor it out: Write the GCF outside the parentheses and divide each term by the GCF.
• This step will often reveal simpler terms and ease further factoring.

Understanding and applying the GCF is essential for simplifying polynomials and solving higher level algebraic equations.