Problem 71 - 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. $$2 n^{3}+6 n^{2}+10 n$$

Step-by-Step Solution

Answer

The polynomial is factored as \(2n(n^2 + 3n + 5)\), not further factorable over integers.

Step 1: Factor Out the Greatest Common Factor (GCF)

First, identify the greatest common factor of all terms in the polynomial. The terms are \(2n^3\), \(6n^2\), and \(10n\). The greatest common factor is \(2n\). Factor \(2n\) out of the polynomial:\[2n(n^2 + 3n + 5)\]

Step 2: Check for Further Factoring

Examine the quadratic expression \(n^2 + 3n + 5\) to see if it can be factored further. Look for two numbers that multiply to 5 (the constant term) and add to 3 (the coefficient of \(n\)). Since such numbers do not exist among integers, \(n^2 + 3n + 5\) is not factorable over the integers. Thus, the polynomial is fully factored.

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) can be thought of as the biggest factor that divides all terms in a polynomial without leaving a remainder. It's like finding the largest size chunk that can be evenly taken out of every piece. For the terms given in our problem—\(2n^3\), \(6n^2\), and \(10n\)—we need to find what can be taken out of all of them evenly.

• For the numerical coefficients (2, 6, and 10), the largest number that divides all three is 2.
• For the variable \(n\), we take the smallest power present in each term, which is \(n\).

Thus, the GCF of these terms is \(2n\). Once identified, the GCF can be "factored out" of the polynomial, simplifying it and making it easier to work with.

Quadratic Expressions

Quadratic expressions appear frequently in math and are of the form \(ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are constants. They are an essential part of algebra and solving them can help us find where a function hits the x-axis. In our problem, after factoring out the GCF, we're left with a quadratic expression: \(n^2 + 3n + 5\).To factor a quadratic expression, we're typically looking for two numbers that, when multiplied together, equal the constant term (in this case, 5), and when added together, equal the linear coefficient (in this case, 3). However, when we can't find such numbers using integer factorization and the expression can't be rearranged further into linear factors, it means the quadratic is prime over the integers. In simpler terms: it’s as simplified as it can get without involving more complex numbers or fractions.

Integer Factorization

Integer factorization in polynomial expressions is about breaking down an expression into products of simpler terms, and ideally into linear factors. This process helps solve equations or simplify expressions by making them more manageable.When we factor polynomials, we generally prefer integer factors since they are straightforward to compute and interpret. In our problem, the quadratic \(n^2 + 3n + 5\) couldn't be factored into smaller expressions involving integers—the numbers just don't work out to balance the multiplication and addition required for factorization.This does not mean the expression can’t be factored differently, but for our purpose of integer factorization, we state it is not possible beyond pulling out the common factors and acknowledging its current form as its simplest over the integers.Knowing when a polynomial is fully factored with integers is crucial in math because it indicates the conclusion of the simplification process, allowing you to work with the expression effectively in equations or graphs.