Problem 57 - College Algebra and Calculus: An Applied Approach

Question

Completely factor the expression.\(-2 x^{2}-4 x+2 x^{3}\)

Step-by-Step Solution

Answer

The fully factored form of the given expression is \(2x(x-2)(x+1)\)

Step 1: Rearrange the expression

Rearrange the expression in descending order of powers of x: \(2x^{3}-2x^{2}-4x\).

Step 2: Identify common factors

Review each term in the polynomial to identify common factors that each term shares. The common factor in this case is 2x.

Step 3: Factor out the common factor

Factor out the common factor 2x from each term: \(2x(x^{2}-x-2)\).

Step 4: Factor the quadratic expression

Factor the quadratic expression \(x^{2}-x-2\). This can be factored further into (x-2)(x+1).

Step 5: Write the fully factored expression

Write out the fully factored expression by combining the factor from step 3 and the factors from step 4: \(2x(x-2)(x+1)\)

Key Concepts

Polynomial Rearrangement

Understanding polynomial rearrangement is central when you start factoring polynomials. It helps to order the chaos that you might first encounter. Imagine you're given scrambled pieces of a puzzle - your task is to put them together to see the whole picture. That's what polynomial rearrangement is about: ordering terms from highest to lowest degree.

As seen in the exercise, rearranging the expression (-2x^2 - 4x + 2x^3) into descending order gives us (2x^3 - 2x^2 - 4x). This reordering is vital because it lays out the polynomial in a standard form, which can make the subsequent steps of identifying common factors and factoring the expression much clearer.

With polynomials neatly arranged, it's easier to observe patterns in the coefficients and the powers of the variable (in this case, 'x'), which is an essential skill in algebra.

Common Factors

When factoring polynomials, one of the first steps is to look for common factors. Spotting commonalities among the terms can simplify the expression and is similar to finding what ingredients are shared in different recipes. Here, in each term of the polynomial (2x^3 - 2x^2 - 4x), we can spot straight away that each term can be divided by 2x.

This shared factor is like a thread tying all the terms together, which allows us to 'pull out' this common piece and thus simplifying the expression further. When we factor out the 2x, we're left with (2x)(x^2 - x - 2). Not only does this look neater, but it also sets the stage for easier further factoring, highlighting the importance of meticulously checking for and factoring out common factors.

Quadratic Expressions

Dealing with quadratic expressions comes up often in algebra, and it's a key concept when factoring polynomials. A quadratic expression is a polynomial of the second degree, which means its highest exponent is two, typically taking the form (ax^2 + bx + c).

In our exercise, after factoring out the 2x, we were left with a quadratic expression (x^2 - x - 2). To factor this further, we look for two numbers that multiply to give us the constant term (-2, in this case) and add to give us the coefficient of the 'x' term (-1, in this case). Discovering these two numbers can sometimes feel like solving a small riddle.

Once we find that the numbers are -2 and 1, we can rewrite the quadratic expression as two binomials: (x - 2)(x + 1). This breakdown into binomials is crucial because it unveils the roots of the quadratic equation, providing deep insights into the function's behavior, such as its graph and its intersections with the x-axis.