Problem 68 - Introductory Algebra for College Students

Question

Factor completely. $$3 x^{3}+14 x^{2}+8 x$$

Step-by-Step Solution

Answer

The completely factored form of the expression \(3x^3 + 14x^2 + 8x\) is \(x(3x + 2)(x + 4)\).

Step 1: Identify the Greatest Common Factor

The first step is to identify and factor out the greatest common factor. Looking at the terms \(3x^3\), \(14x^2\) and \(8x\), the common factor is 'x'. This results in \(x(3x^2 + 14x + 8)\).

Step 2: Factor the Quadratic Expression

Next, factor the quadratic expression inside the parentheses i.e., \(3x^2 +14x +8\). The factors of \(3x^2 +14x +8\) are \((3x+2)\) and \((x+4)\) because (3x+2)(x+4) equals \(3x^2 +14x +8\). So, the expression becomes \(x(3x + 2)(x + 4)\).

Key Concepts

Greatest Common Factor

Understanding the greatest common factor (GCF) is crucial when working with polynomials. The GCF is the largest factor that divides two or more terms without a remainder. To factor a polynomial, it’s essential to first pull out the GCF. This simplifies the polynomial, making the next steps easier.

• For instance, in the polynomial expression \(3x^3 + 14x^2 + 8x\), we examine each term: \(3x^3\), \(14x^2\), and \(8x\).
• The common factor here is \(x\), since \(x\) is present in all terms.
• Factoring out the GCF \(x\), the expression simplifies to \(x(3x^2 + 14x + 8)\).

Remember, recognizing and factoring out the GCF is the first step in simplifying polynomial expressions. It makes the remaining expression simpler, setting the stage for further factorization.

Quadratic Expression

Once the GCF is factored out, you may notice a quadratic expression remaining in the polynomial. A quadratic expression is generally in the form \(ax^2 + bx + c\). Here, the task is to convert this into a product of two binomials.

• In our example, after factoring out the GCF, we have \(3x^2 + 14x + 8\).
• To factor a quadratic expression, look for two numbers that multiply to give the product of \(a\) and \(c\), and add to give \(b\).
• For \(3x^2 + 14x + 8\), the paired numbers are found to be \(2\) and \(12x\).

These numbers help break down the middle term; however, the simpler approach would factor the expression as \((3x + 2)(x + 4)\). Understanding this step is crucial for successful polynomial factorization.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler, non-reducible parts called factors. This process can make solving equations and simplifying expressions much easier.

• After facting out the GCF and dealing with the quadratic expression separately, you would combine these to find the complete factorization of the polynomial.
• In the exercise, the expression \(3x^3 + 14x^2 + 8x\) is fully factored to \(x(3x + 2)(x + 4)\).
• This not only simplifies the polynomial but also makes it easier to solve equations involving it by setting each factor equal to zero.

Grasping factoring as a method depends on understanding how to identify common factors, recognize quadratic patterns, and apply the appropriate techniques to break down larger expressions into manageable pieces. The aim is to learn to see polynomials as products of their factors, making them simpler to work with.