Problem 59 - Introductory Algebra for College Students

Question

Factor completely. $$4 x^{2}+26 x+30$$

Step-by-Step Solution

Answer

The quadratic \(4 x^{2}+26 x+30\) completely factors to \( (2x+5)(2x+6) \)

Step 1: Set up the quadratic

We have a quadratic of the form \( ax^2 + bx + c \) where \( a = 4 \), \( b = 26 \), and \( c = 30 \). Our aim is to find two numbers whose product is \( ac = 4*30 =120 \) and sum is \( b = 26 \).

Step 2: Identify the numbers

The numbers that satisfy these conditions are 10 and 12 since \( 10*12 = 120 \) and \( 10+12 = 22 \).

Step 3: Rewrite the equation

We use these two numbers to rewrite the middle term of the equation. The equation becomes: \( 4x^2 + 10x + 16x + 30 \).

Step 4: Factor by grouping

Arrange the terms so that we can factor by grouping. The equation can be rewritten as \( (4x^2 +10x) + (16x + 30) \). Factoring gives us \( 2x(2x+5) + 2(8x+15) \).

Step 5: Checking and final result

We notice that \( (2x+5) \) is a common factor. Factoring out \( (2x+5) \) gives us the final factored form of the equation: \( (2x+5)(2x+6) \) as the final answer.

Key Concepts

Factoring by Grouping

Factoring by grouping is a strategy used to simplify polynomials by organizing terms into smaller groups that can be factored separately. This method proves particularly valuable when dealing with four-term polynomials or when a common factor is not immediately apparent.
In this process, you start by dividing your polynomial into two groups. Each group should have two terms. In our exercise, the quadratic expression was rewritten as four terms: \( 4x^2 + 10x + 16x + 30 \).
Once grouped, factor out the greatest common factor (GCF) from each pair.
For example:

• First group: \( 4x^2 + 10x \) can be factored as \( 2x(2x + 5) \).
• Second group: \( 16x + 30 \) simplifies to \( 2(8x + 15) \).

After factoring each group, you notice if there is a common factor remaining. Here, the expression contained \( (2x + 5)(2x + 6) \), which indicated that \( (2x + 5) \) is a shared factor, leading us to the final expression.

Quadratic Equations

Quadratic equations are expressions of the form \( ax^2 + bx + c = 0 \). These equations involve a term in \( x^2 \) and typically represent parabolas when graphed. The structure is particularly known for its application in many areas, from physics to finance.
In the given problem, the quadratic expression is \( 4x^2 + 26x + 30 \). Here, \( a = 4 \), \( b = 26 \), and \( c = 30 \). Understanding these components is key to managing quadratic equations effectively.
Quadratic equations can be solved in several ways, such as:

• Factoring
• Completing the square
• Quadratic formula

Factoring is often preferred for equations that allow it, as it provides a straightforward approach, especially when the solutions are rational or integer values.

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as the product of simpler polynomials. It's a useful skill in algebra for simplifying expressions and solving equations, especially when dealing with higher-degree polynomials.
To factor a polynomial like \( 4x^2 + 26x + 30 \), first identify the general form and calculate the product \( ac \) of the first and last coefficients. In this case, it's \( 120 \). The goal is to find two numbers that multiply to \( 120 \) and add to the middle coefficient \( b = 26 \). The numbers \( 10 \) and \( 12 \) meet these requirements.
This step helps in rewriting and breaking down the polynomial into groups, enabling the factoring by grouping method. Successfully factoring a polynomial requires careful attention to detail and practice, but it's an indispensable tool for handling complex algebraic expressions.