Problem 26 - Elementary and Intermediate Algebra

Question

Factor. See Example 1 or Example 6. $$ 5 n^{2}+12 n+7 $$

Step-by-Step Solution

Answer

The factored form of \(5n^2 + 12n + 7\) is \((5n + 7)(n + 1)\).

Step 1: Identify the Coefficients

The given quadratic expression is \(5n^2 + 12n + 7\). Identify the coefficients of the expression: \(a = 5\), \(b = 12\), and \(c = 7\).

Step 2: Calculate Product and Sum

To factor by grouping, calculate the product \(ac = 5 \times 7 = 35\). We need two numbers that multiply to \(35\) and add to \(12\), the middle coefficient.

Step 3: Find Suitable Pair of Numbers

The numbers 5 and 7 multiply to 35 and add to 12. These are the numbers we'll use to rewrite the middle term.

Step 4: Break Middle Term

Rewrite the expression by breaking the middle term 12n into \(5n + 7n\): \(5n^2 + 5n + 7n + 7\).

Step 5: Factor by Grouping

Group the terms and factor each group:\((5n^2 + 5n) + (7n + 7)\).Factor out the greatest common factor from each group: \(5n(n + 1) + 7(n + 1)\).

Step 6: Extract the Common Factor

Notice the common factor \((n + 1)\), factor it out:\((5n + 7)(n + 1)\).

Step 7: Final Step: Verify the Factorization

Expand \((5n + 7)(n + 1)\) to ensure it matches the original expression. Solve:\(5n(n + 1) + 7(n + 1) = 5n^2 + 5n + 7n + 7 = 5n^2 + 12n + 7\). It matches the original expression.

Key Concepts

Polynomial Expression

When working with polynomial expressions, you'll often encounter terms raised to different powers. In a quadratic expression like \(5n^2 + 12n + 7\), the highest power of the variable, here \(n\), is 2.

• The expression has three terms, known as a trinomial.
• Each term has a coefficient: a constant that multiplies the variable raised to a power. Here, 5, 12, and 7 are the coefficients.
• The primary goal when factoring is to rewrite the expression as a product of simpler terms.

Understanding how to identify and work with these terms sets a strong foundation for factoring.

Grouping Method

The grouping method is a technique used to factor polynomial expressions, especially when other methods are not straightforward. This involves rewriting part of the expression to find common groups and simplify.

• Initially, calculate the product of the leading coefficient \(a\) and the constant term \(c\).
• Look for a pair of factors of this product that also sum to the middle coefficient \(b\).
• The expression is then restructured to group terms with common factors.

This strategy is particularly useful when dealing with quadratic expressions, and it helps to simplify the process of factoring by identifying commonalities.

Coefficients Identification

Identifying coefficients is fundamental in factoring. The coefficients are the numbers in front of variables in each term of a polynomial expression.

• The leading coefficient ('\(a\)' in standard form \(ax^2 + bx + c\)) influences the expression's shape and is a key player in the first step of factoring.
• The middle coefficient ('\(b\)' typically links the variable without a power.)
• The constant term ('\(c\)' affects the expression's intercept with the y-axis on a graph.)

By clearly identifying these coefficients, you equip yourself to perform calculations necessary for the grouping method and other factoring strategies.

Factoring by Grouping

Factoring by grouping involves creating smaller groups within the expression that can be independently factored and then recombined.Here's how it generally works:

• Rewrite the middle term using the numbers obtained from the product-sum pairing (in this case, break \(12n\) into \(5n + 7n\)).
• Regroup the terms to form two sets: \((5n^2 + 5n)\) and \((7n + 7)\).
• Factor out the greatest common factor from each group, which simplifies each part.
• Finally, extract the overall common factor, resulting in a simplified product: \((5n + 7)(n + 1)\).

This method is powerful for simplifying complex expressions into manageable parts, facilitating easier calculations and deeper understanding.