Problem 59 - Elementary Algebra

Question

Factor completely. $$ 8 x 3-4 x 2+20 x $$

Step-by-Step Solution

Answer

The polynomial is factored completely as \(4x(2x^2 - x + 5)\).

Step 1: Identify the Greatest Common Factor (GCF)

To begin factoring, we first need to identify the GCF of all the terms in the polynomial: - The terms of the polynomial are: \(8x^3\), \(-4x^2\), and \(20x\).- The coefficients of the terms are 8, -4, and 20, with the common factor being 4.- The variable 'x' is common in all terms, with the smallest power being \(x\).- Thus, the GCF is \(4x\).

Step 2: Factor out the GCF from the Polynomial

We now factor out the GCF (\(4x\)) from each term in the polynomial:- Divide each term by \(4x\): - \(\frac{8x^3}{4x} = 2x^2\) - \(\frac{-4x^2}{4x} = -x\) - \(\frac{20x}{4x} = 5\)- The expression becomes: \(4x(2x^2 - x + 5)\).

Step 3: Check for Further Factoring

Examine the resulting expression \(2x^2 - x + 5\) to see if it can be factored further:- Check if it is a trinomial that can be factored.- Try finding two numbers that multiply to \(2 \times 5 = 10\) and add to -1 (the coefficient of \(x\)).- No such numbers exist, and hence \(2x^2 - x + 5\) cannot be factored further.Thus, \(4x(2x^2 - x + 5)\) is the completely factored form.

Key Concepts

greatest common factor

In algebra, the greatest common factor (GCF) plays a crucial role in simplifying expressions and solving equations. The GCF is the largest factor that divides each of the terms in a polynomial evenly. Finding the GCF is usually the first step in the process of factoring.

When dealing with polynomials, identifying the GCF involves two parts: the numerical coefficients and the variables.

• First, look at the coefficients of each term. In our example, these are 8, -4, and 20. The greatest common factor among these numbers is 4.
• Next, examine the variable parts of each term. Since each term contains an 'x', it becomes part of the GCF. The lowest power of x across the terms is x to the power of 1, thus the GCF includes x itself.

By combining these observations, we determine that the GCF for this polynomial is \(4x\). Taking out the GCF from each term helps set the stage for further simplification and solutions.

algebraic expressions

Algebraic expressions are fundamental in algebra and mathematics as a whole. They consist of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. Understanding the structure of algebraic expressions lays the groundwork for factoring and other operations.

In the exercise example, the expression \(8x^3 - 4x^2 + 20x\) is an algebraic expression we want to simplify by factoring. Each term in this expression has:

• A coefficient: the numerical value which in this case are 8, -4, and 20.
• A variable: represented by 'x', which appears in every term.
• A degree: which is the exponent of ‘x’ and varies from term to term.

Breaking down an algebraic expression into its components allows us to better comprehend the expression's behavior and potential for simplification. When simplifying or factoring, understanding these basics is key.

factoring steps

Factoring polynomials is a process of breaking down a polynomial into its simpler components or factors. This involves several key steps that make it systematic and manageable. The goal is often to express the polynomial as a product of its factors to either simplify it or solve an equation.

Here's a quick guide to factoring using our example:

• Step 1: Identify the GCF:
  First, determine the greatest common factor of all terms. Here, it's \(4x\). By factoring out \(4x\), we simplify the polynomial into more manageable pieces.
• Step 2: Factor Out the GCF:
  Divide each term in the polynomial by the GCF to extract it. After factoring out \(4x\), the polynomial becomes \(4x(2x^2 - x + 5)\).
• Step 3: Check for Further Factoring:
  Finally, always check if the remaining polynomial can be factored further. This involves assessing if it's a simple trinomial or another factorable form. In our case, \(2x^2 - x + 5\) is not factorable further.

Following these steps ensures you cover all possibilities and achieve a completely factored form, essential in simplifying expressions and solving equations efficiently.