Problem 91 - Algebra for College Students

Question

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$2 n^{3}+14 n^{2}-20 n$$

Step-by-Step Solution

Answer

The polynomial \(2n^3 + 14n^2 - 20n\) factors to \(2n(n + 10)(n - 1)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, look at the polynomial \(2n^3 + 14n^2 - 20n\). Identify the greatest common factor of all the terms, which is \(2n\). Factoring \(2n\) out, we get: \[2n(n^2 + 7n - 10)\].

Step 2: Factor the Quadratic Expression

Focus on the quadratic expression \(n^2 + 7n - 10\). We need two numbers that multiply to \(-10\) and add to \(7\). The numbers that satisfy this are \(10\) and \(-1\).

Step 3: Apply the Factoring Technique

Rewrite \(n^2 + 7n - 10\) as \((n + 10)(n - 1)\). By substituting, the factored form of the polynomial becomes: \[2n(n + 10)(n - 1)\].

Key Concepts

Understanding the Greatest Common Factor (GCF)

When faced with factoring a polynomial, the first step is often to identify the greatest common factor (GCF). The GCF is the largest expression that divides all terms of the polynomial without leaving a remainder.

To find the GCF in a polynomial like \(2n^3 + 14n^2 - 20n\), look for the largest number and the highest power of any variable that appears in every term.

• The coefficient part: Here, we look at the numbers 2, 14, and 20. The greatest number that divides all three is 2.
• The variable part: Since each term includes an \(n\), and the smallest exponent is 1, the GCF of the variable part is \(n\).

Thus, the GCF of the polynomial is \(2n\). Factoring it out simplifies the polynomial to \(2n(n^2 + 7n - 10)\). Understanding the GCF makes the task of factoring more manageable, providing a simpler expression to work with going forward.

What Are Quadratic Expressions?

Quadratic expressions are polynomials of degree two, typically in the form \(ax^2 + bx + c\). They can often be factored into two binomials.

In the expression \(n^2 + 7n - 10\), \(a = 1\), \(b = 7\), and \(c = -10\). Our goal is to break this down into a product of two simpler expressions, which we will represent as \((n+p)(n+q)\).

Here’s how we do it:

• We need numbers \(p\) and \(q\) such that \(p \times q = c = -10\).
• Also, \(p + q = b = 7\).

Finding such numbers can be tricky, but in this example, they are 10 and -1. So, \(n^2 + 7n - 10\) can be expressed as \((n+10)(n-1)\). Understanding the structure of quadratic expressions is crucial for effective factoring.

Mastering Factoring Techniques

Factoring techniques are essential skills in algebra, allowing one to simplify expressions and solve equations more effectively.

The original exercise used a combination of techniques to transform \(2n^3 + 14n^2 - 20n\) into its factored form. Here's a breakdown:

• First, apply the GCF technique by factoring out the \(2n\), simplifying the expression to \(2n(n^2 + 7n - 10)\).
• Next, employ the "factoring by grouping" technique on the quadratic \(n^2 + 7n - 10\), breaking it down into \((n+10)(n-1)\).

The complete factored form of the polynomial is \(2n(n+10)(n-1)\). Each step in the process demands attention to detail, but by learning these techniques, you can tackle a wide range of polynomial factoring problems with confidence. Practicing these strategies helps in recognizing patterns and applying them effectively in different scenarios.