Problem 42 - Algebra 2

Question

Factor completely. If the polynomial is not factorable, write prime. $$ 3 n^{2}+21 n-24 $$

Step-by-Step Solution

Answer

The completely factored form is \(3(n+8)(n-1)\).

Step 1: Identify the Common Factor

First, look for a common factor in all the terms of the polynomial \(3n^2 + 21n - 24\). Notice that each term is divisible by 3. Factoring out the 3 gives: \[ 3(n^2 + 7n - 8) \]

Step 2: Factor the Quadratic Expression

Next, focus on the quadratic expression \(n^2 + 7n - 8\). We need two numbers that multiply to -8 and add to 7. These numbers are 8 and -1. Therefore, we can rewrite the quadratic as \((n+8)(n-1)\).

Step 3: Combine All the Factors

Bring the common factor and the factored quadratic expression together. The complete factorization of the polynomial is: \[ 3(n+8)(n-1) \]

Key Concepts

Identifying Common Factors

Recognizing and extracting the common factors is a fundamental step in factoring polynomials. A common factor is a number or expression that divides all terms in the polynomial without leaving a remainder.
In the polynomial given in our exercise, which is \(3n^2 + 21n - 24\), we start by checking each term for any common numerical factor.

• The numbers in the terms are 3, 21, and -24.
• All of these numbers can be evenly divided by 3.

Hence, 3 is the common factor for this polynomial.
By factoring out 3, we simplify our expression to \(3(n^2 + 7n - 8)\). Identifying and factoring out common factors simplifies the polynomial, making further operations much easier.

Understanding Quadratic Expressions

Quadratic expressions are polynomials of degree two, meaning the highest exponent is 2. They generally take the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.
In our scenario, we are dealing with the quadratic expression \(n^2 + 7n - 8\), extracted from the original polynomial.
The goal is to express it as a product of two binomials.

• Identify two numbers that multiply to the constant term, which is -8,
• and add up to the coefficient of the linear term, which is 7.

This concept is crucial because it helps convert complex quadratics into simpler, factorable expressions.
The correct pairing here is 8 and -1, satisfying both multiplication and addition requirements. Therefore, the quadratic \(n^2 + 7n - 8\) factors into \((n+8)(n-1)\). Mastering these relationships in quadratics increases your ability to handle more advanced polynomial equations.

Steps to Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of simpler polynomials. It is an essential skill for solving equations and simplifying expressions. Here’s a simple way to approach factorization:
1. **Identify Common Factors:** Start by looking for any common factor among the coefficients and factor them out, just like we extracted 3 in our previous example.
2. **Focus on Quadratics:** Once common factors are out, focus on two-term (binomial) or three-term (trinomial) expressions. For quadratics, look for two numbers that relate as discussed in quadratic expressions, like finding numbers that multiply and add to specific values.
3. **Recombine the Factors:** After breaking down into simpler terms, like dividing \(n^2 + 7n - 8\) into \((n+8)(n-1)\), don’t forget to include any previously factored common terms. Combine them for the complete factorization.

• This step-by-step ensures no factor is missed and the polynomial is fully reduced to its simplest form.

Mastery of polynomial factorization not only helps with solving equations more efficiently but also enhances your general mathematical problem-solving skills.