Problem 73 - Introductory Algebra for College Students

Question

Use the negative of the greatest common factor to factor completely. $$-2 x^{3}-6 x^{2}+8 x$$

Step-by-Step Solution

Answer

The completely factored form of the equation \(-2x^{3}-6x^{2}+8x\) is \(-2x(x-1)(x-4)\).

Step 1: Identify the Greatest Common Factor

Determine the greatest common factor (GCF) of all the terms in the equation, which seems to be \(2x\).

Step 2: Extract the GCF

Divide the equation by the GCF to simplify the equation. Doing so, the equation becomes: \(-x^{2}-3x+4\). Note: we've extracted the negative of the GCF, as required by the exercise.

Step 3: Factor the Quadratic Equation

Next step is to factor the quadratic equation \(-x^{2}-3x+4\). In this case, it factors into \(-(x-1)(x-4)\), since \(1 * 4 = 4\) and \(1 + 4 = 5\) which coincides with the coefficients of the terms.

Key Concepts

Greatest Common Factor (GCF)

The greatest common factor (GCF) is a concept in mathematics that is essential when simplifying algebraic expressions. It refers to the largest factor that divides all terms in an expression without leaving a remainder. When dealing with polynomials, identifying the GCF allows you to factor it out, simplifying the expression significantly.

To find the GCF of terms in a polynomial:

• Identify the numerical parts and variable parts separately.
• For numbers, find the highest common number that divides each coefficient.
• For variables with exponents, take the variable with the smallest power common to all terms.

In our example, the terms are \(-2x^3\), \(-6x^2\), and \(8x\). By analyzing the numerical parts, the GCF of \(2, 6, \text{and} \ 8\) is \(2\). For the variable \(x\), the smallest exponent is \(1\), which occurs in all terms, making \(x\) part of the GCF. Therefore, the GCF of the expression is \(2x\), and using its negative refines the factoring process.

Quadratic Equations

Quadratic equations are a crucial topic in algebra. They usually appear in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The highest degree of the variable, which is \(x^2\) in this case, distinguishes it as a quadratic equation.

The solutions to quadratic equations can often be found by factoring, using the quadratic formula, or completing the square. When factoring, you aim to express the quadratic equation as a product of two binomials. However, this is only possible when the equation fits certain patterns or when the numbers related to the equation, such as coefficients, have a straightforward factorization.

In the given exercise, after factoring out the negative GCF from the original expression, we encountered a quadratic, \(-x^2 - 3x + 4\). To factor this, observe how the term pairs 1 and 4 work together, using the trial and error method for matching coefficients to split the middle term logically, which results in \(-(x - 1)(x - 4)\). This showcases the connection between the roots and the factors of the quadratic.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the building blocks of algebra, enabling us to construct equations, inequalities, and functions.

Working with algebraic expressions often involves simplifying or factoring to find solutions or to reorganize them into simpler forms. Simplification involves combining like terms, which are terms with the same variable raised to the same power, and reducing expressions by factoring.

In our exercise, we begin with the expression \(-2x^3 - 6x^2 + 8x\). Applying the principles of factoring polynomials to algebraic expressions:

• First, identify the GCF, which helps to simplify by factoring out common elements.
• Then, restructure the expression into multiple simpler expressions or products of expressions.

These steps lead us to see how the original complex algebraic expression can be rewritten into a more advantageous form, revealing the expression's true characteristics and simplifying calculations greatly.