Question

Factor. $$12 x^{3}+20 x^{2}+32 x$$

Short answer

Answer: The factored form of the expression is \(4x(3x^2 + 5x + 8)\).

Step-by-step solution

Identify the Greatest Common Factor (GCF)

The first step to factor the given expression is to determine the greatest common factor (GCF) for the coefficients and the variables in the expression. We have: Coefficient GCF: GCF(12, 20, 32) = 4 Variable GCF: GCF(\(x^3, x^2, x\)) = \(x\) The GCF is \(4x\).

Divide each term by the GCF

Now, divide each term of the expression by the GCF we found in Step 1. $$\frac{12x^3}{4x} + \frac{20x^2}{4x} + \frac{32x}{4x}$$ This simplifies to: $$3x^2 + 5x + 8$$

Factor the GCF out of the expression

Finally, factor the GCF out of the original expression by multiplying it by the simplified expression from Step 2: $$4x(3x^2 + 5x + 8)$$ The factored expression is $$4x(3x^2 + 5x + 8)$$

Key concepts

Greatest Common Factor (GCF)

To factor an algebraic expression successfully, you first need to find the Greatest Common Factor (GCF). Think of the GCF as the largest factor that divides each term in the expression equally. By finding the GCF, we simplify the problem and make it easier to work with individual parts of the expression.

Let's break down this process:

• Identify the GCF of the coefficients (numerical parts) of the terms. In our example with the expression \(12x^3 + 20x^2 + 32x\), we determine the GCF of 12, 20, and 32, which is 4.
• Next, find the GCF of the variable parts. For \(x^3, x^2,\) and \(x\), the smallest power that divides all is \(x\). Hence, the GCF of the variables is \(x\).

The overall GCF of the expression is the product of these two GCFs, which is \(4x\). Finding the GCF is crucial as it helps to simplify algebraic expressions, turning them into more manageable forms.

Polynomial Factoring

Once we have the GCF, the next step is factoring the polynomial. This process involves restructuring the expression by removing the GCF from each term, leaving a simpler expression inside the brackets. This is akin to reversing the distributive property.

Using our expression \(12x^3 + 20x^2 + 32x\), after determining that the GCF is \(4x\), we divide each term by \(4x\):

• \(\frac{12x^3}{4x} = 3x^2\)
• \(\frac{20x^2}{4x} = 5x\)
• \(\frac{32x}{4x} = 8\)

Putting it all together, the expression \(12x^3 + 20x^2 + 32x\) becomes \(4x(3x^2 + 5x + 8)\) when the GCF is factored out. This compact form not only simplifies the expression but also prepares it for further operations or applications.

Simplifying Expressions

Simplifying expressions is the fundamental cherry on top of factoring. Once we've factored the polynomial by removing the GCF, we often need to simplify further by evaluating or tidying up the expression, making it as clear and concise as possible.

Simplifying is pivotal because:

• It makes mathematical expressions easier to understand and work with.
• It helps in solving algebraic equations more efficiently.
• It facilitates the identification of common patterns or solutions in complex algebraic problems.

Using our example, the simplified formulation \(4x(3x^2 + 5x + 8)\) is straightforward and ready for use, further manipulation, or even solving if it were an equation. Being proficient at simplifying will boost your confidence in tackling a variety of algebraic problems.