Problem 35 - Elementary and Intermediate Algebra

Question

Factor. $$ 36 x^{3}+12 x^{2}+x $$

Step-by-Step Solution

Answer

The expression factors to \(x(6x + 1)^2\).

Step 1: Identify the Greatest Common Factor (GCF)

To begin factoring, identify the greatest common factor (GCF) of all the terms in the expression. Here, each term contains the variable \(x\). Also, look at the numerical coefficients: 36, 12, and 1. The GCF of these coefficients is 1. Therefore, our GCF is simply \(x\). Factor \(x\) out of each term.

Step 2: Factor out the GCF

Factor \(x\) from each term in the expression: \[ 36x^3 + 12x^2 + x = x(36x^2 + 12x + 1) \] Now the expression inside the parenthesis, \(36x^2 + 12x + 1\), needs further factoring.

Step 3: Factor the Trinomial

Focus on factoring the quadratic trinomial: \(36x^2 + 12x + 1\). Look for two numbers that multiply to \(36 \times 1 = 36\) and add to \(12\). These numbers are \(6\) and \(6\).

Step 4: Apply the Factoring Technique

Since \(6\) and \(6\) multiply to \(36\) and add to \(12\), rewrite \(12x\) as \(6x + 6x\): \[ x(36x^2 + 6x + 6x + 1) \] Group the terms: \[ x((36x^2 + 6x) + (6x + 1)) \]

Step 5: Factor by Grouping

Now factor the common elements from each group: \[ x(6x(6x + 1) + 1(6x + 1)) \] Since \(6x + 1\) is common in both groups, factor it out: \[ x((6x + 1)(6x + 1)) \] This can be simplified to: \[ x(6x + 1)^2 \]

Step 6: Write the Final Factored Form

Combining the results, the completely factored form of the initial expression is: \[ x(6x + 1)^2 \]

Key Concepts

Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest factor common to all terms in an algebraic expression. When factoring polynomials, identifying the GCF is a pivotal first step. To find it, examine both the numerical coefficients and the variable parts. For example, in an expression like

• 36x^3
• 12x^2
• x

the GCF isn't just among the numbers (36, 12, and 1). It's also about ensuring we incorporate any common variables. Here, each term at least contains a single factor of \(x\). Once identified, the GCF (in this case, \(x\)) is factored out, simplifying the polynomial for further operations. This simplification lays the groundwork for more complex factoring, such as trinomial factoring or grouping, by reducing the polynomial into a simpler form. Remember, the goal of finding the GCF is to reduce the expression to its most basic components to ease further analysis and factorization.

Quadratic Trinomial

A quadratic trinomial is a polynomial with three terms, generally written in the form \(ax^2 + bx + c\). It's called 'quadratic' because the highest degree (or power) of the variable is two. Understanding how to factor these trinomials is crucial, as they frequently appear in algebra.

The goal in factoring a quadratic trinomial is to express it as a product of two binomials. This involves finding two numbers whose

• product equals the coefficient of \(a\) (here it's multiplied by \(c\))
• sum equals the middle term's coefficient \(b\)

Once these two numbers are found, the middle term can be split or rewritten, allowing an easier application of the factor-by-grouping technique. It's like breaking down the problem into smaller, more manageable parts. This will help to create a factorable expression. Breaking down \(bx\) using these numbers enables you to group and factor with newfound simplicity.

Factoring by Grouping

Factoring by grouping is a versatile method when dealing with four terms in a polynomial, such as after rewriting a quadratic trinomial. The principle is straightforward: group the terms so that each subset can be factored out, then look for a common factor to bring it all together.

Start by identifying pairs of terms with common factors you can "group" together like this,

• You first create two separate binomials from one polynomial.
• Each pair is then factored individually.
• Finally, see if the result has a common binomial factor to simplify further.

This technique works beautifully when the middle term of a trinomial is written as a sum of two numbers, enabling the formation of factorable binomials. It's a logical step that ties together like solving a puzzle - finding and connecting parts that successfully yield a complete, factored expression. The result will often surprise you with its simplicity, as complex expressions transform into neat and elegant products of binomials.