Problem 53 - Introductory Algebra for College Students

Question

Factor completely. $$4 x^{3}+12 x^{2}-72 x$$

Step-by-Step Solution

Answer

The expression \(4x^{3} + 12x^{2} - 72x\) factors completely to \(4x(x+6)(x-3)\).

Step 1: Identify the GCF

In the given expression \(4x^{3} + 12x^{2} - 72x\), 4x is a common factor of each term. By applying the rule of distribution in reverse, a common factor can be factored out from each term.

Step 2: Factor out the GCF

By factoring out the common factor, the given expression becomes \(4x(x^{2} + 3x - 18)\) . You simply divide each term of the expression by 4x.

Step 3: Factor the quadratic equation

The quadratic equation \(x^{2} + 3x - 18\) can be further factored by finding two numbers that multiply to -18 and add to 3. The numbers 6 and -3 fulfill these conditions. So the expression becomes \(x^{2} + 3x - 18 = (x+6)(x-3)\)

Step 4: Combine the factored expressions

Combine the factored quadratic expression from Step 3 with the factored out GCF from Step 2. Therefore, the expression becomes \(4x(x+6)(x-3)\).

Key Concepts

Greatest Common Factor (GCF)

Grasping the concept of the Greatest Common Factor (GCF) is essential for simplifying algebraic expressions and facilitating easier operations with them. The GCF is the highest number that evenly divides all the terms in the expression. In the exercise provided, we are given the polynomial 4x^{3} + 12x^{2} - 72x. To find the GCF, look at the coefficients of each term—which are 4, 12, and 72—and identify the largest number that divides all of them, which in this case is 4. Additionally, since each term contains an x, and the smallest power of x is 1, the GCF includes this variable as well, making it 4x.

Factoring out the GCF simplifies complex expressions and makes subsequent factoring steps more manageable. When you factor out the GCF, you are essentially reversing the distribution rule (more on that rule later), breaking down the expression into a product of simpler parts. For example, factoring the GCF from the given polynomial leads us to 4x(x^{2} + 3x - 18), setting the stage for further factoring inside the parentheses.

Factoring Quadratic Equations

Once the GCF has been factored out, it's time to deal with the remaining quadratic equation, which is often in the form ax^{2} + bx + c. In the educational context, factoring quadratic equations is akin to finding two binomials that, when multiplied together, give you the original quadratic equation. To achieve this, one must search for two numbers that multiply to a*c (the coefficient of x^{2} and the constant term) and simultaneously add up to b (the coefficient of x).

In our exercise, we apply this method to x^{2} + 3x - 18. By inspection, the numbers 6 and -3 multiply to -18 and add up to 3. Hence, the quadratic factors into (x+6)(x-3). This process is a cornerstone in algebra because it lays the groundwork for solving quadratic equations and understanding the behavior of quadratic functions. Factoring quadratics is a valuable tool in finding roots or zeroes of a function, integral for graphing parabolas and in other areas of mathematics.

Distribution Rule in Algebra

The distribution rule, also known as the distributive property, is fundamental in algebra. It involves the spreading of a term over a summation or subtraction inside parentheses. More formally, it states that a(b + c) = ab + ac. Essentially, you multiply the term outside the parentheses by each term inside.{% raw %}

{% endraw %}Returning to our original exercise, factoring out the GCF was made possible by understanding the distribution rule in reverse. Once the GCF 4x is factored out, we essentially 'undistribute' it from each term of the expression, leading to simpler components within the parentheses. Understanding the distribution rule is vital for recognizing opportunities to factor expressions and for expanding expressions when necessary. Additionally, it's a crucial concept in more advanced algebra, helping to grasp other properties like the rules for exponents and understanding polynomial multiplication and division.