Problem 167 - Elementary Algebra

Question

Factor. \(5 n^{2}+21 n+4\)

Step-by-Step Solution

Answer

The factorization is \((n + 4)(5n + 1)\).

Step 1: Identify and Setup the Polynomial

The polynomial to factor is \(5n^2 + 21n + 4\).

Step 2: Multiply the Leading Coefficient and Constant

Multiply the leading coefficient (5) by the constant term (4): \(5 \times 4 = 20\).

Step 3: Find Two Numbers that Multiply to the Product and Add to the Middle Coefficient

We need two numbers that multiply to 20 and add to 21. These numbers are 20 and 1.

Step 4: Rewrite the Middle Term

Rewrite the middle term (21n) using the numbers found: \(5n^2 + 20n + n + 4\).

Step 5: Factor by Grouping

Group the terms: \((5n^2 + 20n) + (n + 4)\).

Step 6: Factor Out the Greatest Common Factor (GCF) From Each Group

Factor out the GCF from each group: \(5n(n + 4) + 1(n + 4)\).

Step 7: Factor Out the Common Binomial

The common binomial is \((n + 4)\). Factor it out to get: \((n + 4)(5n + 1)\).

Key Concepts

Polynomial Factoring

Factoring polynomials is an essential skill in algebra. It means breaking down a polynomial into simpler terms (factors) that, when multiplied together, give the original polynomial.

For example, if we start with the polynomial \(5n^2 + 21n + 4\), we can factor it to find expressions that are easier to work with. This process often simplifies equations and helps solve algebraic problems more efficiently.
When factoring polynomials, keep in mind the goal is to find expressions within the polynomial that can be combined, or factored out, to create a product of simpler terms.

• Check for overall common factors.
• Look for patterns like perfect squares or the difference of squares.
• Use specific strategies, such as grouping, to handle complex polynomials.

Grouping Method

The grouping method is a powerful tool for factoring polynomials, especially when dealing with four terms. It involves pairing terms to find and extract the common factors efficiently.

In our polynomial, \(5n^2 + 21n + 4\), we start by rewriting it so that we have four terms that can be grouped. This involves finding two numbers that multiply to the product of the leading coefficient and the constant term and add up to the middle coefficient.
For instance:

• Multiply the leading coefficient (5) by the constant term (4): \(5 \times 4 = 20\).
• Identify two numbers that multiply to 20 and add up to 21. These are 20 and 1.
• Rewrite the polynomial: \(5n^2 + 20n + n + 4\).
• Group the terms to find common factors: \((5n^2 + 20n) + (n + 4)\).

The main aim is to pair terms so common factors can be factored out, simplifying the polynomial into smaller parts that can be managed more effectively.

Greatest Common Factor (GCF)

The Greatest Common Factor, or GCF, is the largest factor shared by terms in a polynomial. Identifying and factoring out the GCF simplifies the polynomial and is often a crucial first step in the factoring process.

In our example, after grouping terms \(5n^2 + 20n + n + 4\) into \((5n^2 + 20n) + (n + 4)\), we look for the GCF in each group:

• The GCF of \(5n^2\) and \(20n\) is \(5n\), so we factor out \(5n\): \(5n(n + 4)\).
• The GCF of \(n\) and \(4\) is 1, so we factor out 1: \(1(n + 4)\).

Now we have \(5n(n + 4) + 1(n + 4)\). Finally, we notice \((n + 4)\) is a common binomial and can be factored out, resulting in: \((n + 4)(5n + 1)\).
Remember, finding the GCF is like finding the hidden pieces of the polynomial puzzle, making complex expressions much easier to work with.