Problem 97 - Algebra for College Students

Question

Consider the following approach to factoring \(12 x^{2}+54 x+60\) $$ \begin{aligned} 12 x^{2}+54 x+60 &=(3 x+6)(4 x+10) \\ &=3(x+2)(2)(2 x+5) \\ &=6(x+2)(2 x+5) \end{aligned} $$ Is this a correct factoring process? Do you have any suggestion for the person using this approach?

Step-by-Step Solution

Answer

The process is correct but could be streamlined by first factoring out the GCF of 6.

Step 1: Understand the expression

We start with the quadratic expression \(12x^2 + 54x + 60\). The goal is to factor this into simpler expressions.

Step 2: Analyze the Provided Solution

The provided solution attempts to factor the expression by rewriting it as \((3x + 6)(4x + 10)\), then further breaking it down to \(3(x+2)(2)(2x+5)\), and collapsing it to \(6(x+2)(2x+5)\).

Step 3: Verify Initial Factoring

Check if \((3x + 6)(4x + 10)\) correctly multiplies to the original expression. Expanding, we have:\(3x \cdot 4x + 3x \cdot 10 + 6 \cdot 4x + 6 \cdot 10 = 12x^2 + 30x + 24x + 60 = 12x^2 + 54x + 60\).This factorization holds true.

Step 4: Evaluate Simplifications

The solution then simplifies \((3x + 6)(4x + 10)\) to \(3(x+2)(2)(2x+5)\). Breaking this down:* \(3(x+2)\) is equivalent to \(3x + 6\).* \(2(2x+5)\) is equivalent to \(4x + 10\).Thus, the simplification from \((3x + 6)(4x + 10)\) to \(3(x+2)(2)(2x+5)\) is valid.

Step 5: Check Final Factorization

The final expression is \(6(x+2)(2x+5)\) which combines the constants 3 and 2 into 6, maintaining the expression's validity.

Step 6: Recommend Correct Factorization Approach

A more straightforward method is using the greatest common factor (GCF). The GCF for \(12x^2 + 54x + 60\) is 6, leading to:\[ 12x^2 + 54x + 60 = 6(2x^2 + 9x + 10) \]Further factor \(2x^2 + 9x + 10\) to get\[ =(2x + 5)(x + 2) \]Thus the entirely factored form becomes \(6(2x + 5)(x + 2)\).

Key Concepts

Quadratic Expressions

Quadratic expressions are a crucial part of algebra, usually taking the form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable. These expressions follow a polynomial degree of two, meaning the highest power of the variable is squared.
Understanding quadratic expressions is essential because they often need to be simplified or factored in algebraic operations.

Factoring refers to expressing the quadratic as a product of simpler polynomials. This process can help in solving quadratic equations or simplifying expressions, making it easier to find solutions or understand the structure of the equation.
A quadratic expression like \(12x^2 + 54x + 60\) can be simplified by finding its factors, helping us see patterns or solutions that aren't immediately obvious.

Greatest Common Factor

The Greatest Common Factor (GCF) is an important concept when working with algebraic expressions. It involves finding the largest factor common to all terms in an expression.
In the context of the expression \(12x^2 + 54x + 60\), the GCF is the largest constant that can divide each of the coefficients. The GCF is crucial because it simplifies the process of factoring by reducing the expression to its simplest form.

To find the GCF for \(12x^2 + 54x + 60\), you need to look at the coefficients: 12, 54, and 60. The largest number that divides all of these is 6.

• This means 6 can be factored out, simplifying the expression considerably to: \[ 6(2x^2 + 9x + 10) \]

Using the GCF as the first step in factoring makes subsequent steps much more straightforward and manageable, providing a clear path to complete the factorization.

Simplification Process

The simplification process in algebra involves reducing complex expressions into simpler, more manageable forms. For quadratic expressions, this often means factoring them into products of binomials.

In the problem provided, the initial expression \((3x+6)(4x+10)\) was correctly simplified into smaller factors. Each step involves breaking down terms and finding expressions that, when multiplied, give back the original equation.

• Starting with \((3x + 6)\), it simplifies further to \(3(x + 2)\).
• Similarly, \((4x + 10)\) simplifies to \(2(2x + 5)\).

By combining these, we reach the product \(6(x+2)(2x+5)\), representing the expression in its factored form, ready for further operations or solving. Simplification is key in making complex problems more understandable and solvable without getting lost in overly complicated terms.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and operations (addition, subtraction, multiplication, division) that represent mathematical relationships. They include terms, which are the individual components separated by addition or subtraction signs.

Expressions like \(12x^2 + 54x + 60\) represent a combination of terms involving the variable \(x\). These expressions function as the building blocks of algebra. They need simplifying into smaller, usable parts for problem-solving or manipulation, particularly in equations.

• Algebraic expressions are versatile and can be added, subtracted, multiplied, divided, or factored to suit different problems or solutions.
• Factoring, specifically, allows you to see the underlying structure of the expression, such as roots or zeros, which are pivotal in solving equations.

Mastering algebraic expressions and their manipulation is foundational to tackling more complex mathematical challenges, providing tools to break down and understand complicated problems more thoroughly.