Problem 46 - Algebra 2

Question

Factor each expression. $$ 3 x^{2}+12 x+9 $$

Step-by-Step Solution

Answer

The factored form of the polynomial \(3x^{2} + 12x + 9\) is \(3(x + 1)(x + 3)\).

Step 1: Factor out the Greatest Common Factor (GCF)

The first step would be to factor out the greatest common factor (GCF) from the original expression. In this case, the GCF for the terms \(3x^{2}\), \(12x\), and \(9\) is 3. Factoring out the GCF, the expression becomes \(3(x^{2} + 4x + 3)\).

Step 2: Factor the Quadratic

Next, factor the quadratic \(x^{2} + 4x + 3\). Be on the lookout for two numbers that add up to 4 and multiply to 3, because that is necessary for it to be factorable. The numbers 1 and 3 fit these conditions. Thus, \(x^{2} + 4x + 3\) can be factored into \((x + 1)(x + 3)\).

Step 3: Write the Final Result

Substitute the factored quadratic back into the equation where the GCF was factored out. Therefore, the expression \(3x^{2} + 12x + 9\) factors to \(3(x + 1)(x + 3)\).

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is a vital concept in algebra that helps us simplify expressions and find their factors.
In the context of quadratic expressions, the GCF is the highest number that divides all terms in the expression without a remainder.
By factoring out the GCF, it becomes easier to see the remaining factors of the expression. When approaching a quadratic expression like \(3x^2 + 12x + 9\), the first task is to identify and factor out the GCF.
Here's how we can do that:

• Look at each coefficient: 3 (from \(3x^2\)), 12 (from \(12x\)), and 9.
• The GCF of 3, 12, and 9 is 3.
• Divide each term in the expression by 3: \(3x^2\div3 = x^2, 12x\div3 = 4x, 9\div3 = 3\).

This process results in the simplified expression: \(3(x^2 + 4x + 3)\). By extracting the GCF first, we make the quadratic expression easier to work with in subsequent steps.

Factoring Techniques

Factoring techniques are strategies used to break down polynomials into simpler components, which can be incredibly useful for solving equations.
One of the most common techniques for quadratics is looking for two numbers that can add up to a specific term and multiply to another. In our example,

• We identified the quadratic \(x^2 + 4x + 3\) from the expression \(3(x^2 + 4x + 3)\).
• To factor it, we searched for two numbers that add up to 4 (the linear coefficient) and multiply to 3 (the constant term).
• The numbers 1 and 3 fit these conditions.

This means the quadratic \(x^2 + 4x + 3\) can be rewritten as \((x + 1)(x + 3)\). Factoring out polynomials can often be seen as solving a puzzle, where these techniques help simplify expressions down to their core components for easier analysis.

Quadratic Expressions

Quadratic expressions are a central part of algebra, characterized by having the highest power of the variable as 2. These expressions generally take the form \(ax^2 + bx + c\). Quadratics are widely encountered in mathematics courses due to their relevance in various applications like physics, engineering, and economics.
To effectively deal with them:

• Identify whether the expression can be factored or requires completing the square.
• Use the quadratic formula when necessary.
• Learn to recognize special forms such as perfect square trinomials or the difference of squares.

In our example, the quadratic \(x^2 + 4x + 3\) already fits a standard form that's easily factorable by guessing and checking.
This characteristic makes understanding quadratic expression techniques essential for both simplifying problems and solving equations efficiently.