Problem 90 - Elementary Algebra

Question

Factor by grouping. $$ 6 x 5-4 x 3-9 x 2+6 $$

Step-by-Step Solution

Answer

The factored expression is \((3x^2 - 2)(2x^3 - 3)\).

Step 1: Grouping Terms

To factor by grouping, we will first group the terms in pairs. The given polynomial is \( 6x^5 - 4x^3 - 9x^2 + 6 \). We will group it as follows: \((6x^5 - 4x^3) + (-9x^2 + 6)\).

Step 2: Factor the First Group

Look at the first group \( 6x^5 - 4x^3 \). Identify the common factor in this group. Here, we can factor out \(2x^3\), resulting in \(2x^3(3x^2 - 2)\).

Step 3: Factor the Second Group

Now, consider the second group \(-9x^2 + 6\). The common factor here is \(-3\), so we can factor it out, resulting in \(-3(3x^2 - 2)\).

Step 4: Combine the Groups

After factoring by grouping, we have \(2x^3(3x^2 - 2) - 3(3x^2 - 2)\). Notice that \((3x^2 - 2)\) is a common factor in both terms.

Step 5: Factor out the Common Factor

Factor out the common factor \((3x^2 - 2)\) from the expression. This gives us \((3x^2 - 2)(2x^3 - 3)\).

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operations. Unlike equations, algebraic expressions do not have an equality sign. They can be as simple as a number, such as 5, or more complex like the polynomial \(6x^5 - 4x^3 - 9x^2 + 6\). Polynomials, in particular, are algebraic expressions that consist of variables raised to whole number powers with coefficients. The degree of the polynomial refers to the highest power of the variable in the expression.
Polynomials can be added, subtracted, multiplied, and divided much like numbers. However, when it comes to simplifying or factoring them, you often need to recognize patterns or use different techniques to break them down. Factoring is the process of writing the polynomial as a product of its factors, making it easier to handle and solve.
When you work with algebraic expressions, understanding the structure or pattern can help immensely in applying the correct method, like factoring by grouping, which is used in the original exercise.

Common Factor

A common factor is a number or expression that divides exactly into two or more numbers or expressions. In the context of algebraic expressions, finding a common factor is crucial when simplifying or factoring expressions. Take the polynomial \(6x^5 - 4x^3\), which you can factor by identifying the greatest common factor in the terms.
In this case, each term is divisible by \(2x^3\), making \(2x^3\) the common factor. Once you factor \(2x^3\) out of the terms, the remaining expression becomes \(3x^2 - 2\). Similarly, in the expression \(-9x^2 + 6\), the common factor \(-3\) is used to simplify this group to \(-3(3x^2 - 2)\).
Recognizing common factors in polynomial expressions not only helps in factoring but also in simplifying the process of solving or evaluating these expressions. It allows you to extract simpler expressions from complex polynomials, making them more manageable.

Grouping Method

The grouping method is a technique used to factor polynomials, especially those with four or more terms. This method involves grouping terms into pairs or sets that have a common factor, and then factoring each group individually.
In the original exercise, the polynomial \(6x^5 - 4x^3 - 9x^2 + 6\) is factored using the grouping method. The polynomial is first broken into two groups: \((6x^5 - 4x^3)\) and \((-9x^2 + 6)\). Each group is factored separately using the common factor identified in each set.

• The first group \(6x^5 - 4x^3\) is factored to \(2x^3(3x^2 - 2)\).
• The second group \(-9x^2 + 6\) is factored to \(-3(3x^2 - 2)\).

After factoring both groups, notice that the expression \((3x^2 - 2)\) is common in both. The final step is to factor this common expression out, resulting in \((3x^2 - 2)(2x^3 - 3)\).
The grouping method simplifies the process of breaking down complex polynomials by breaking them into smaller, more manageable expressions. This method is particularly helpful when dealing with higher-degree polynomials.