Problem 15 - Elementary Algebra

Question

For the following problems, factor the polynomials. $$ 12 x^{2}+18 x $$

Step-by-Step Solution

Answer

Answer: The factored form of the polynomial \(12x^2 + 18x\) is \(6x(2x + 3)\).

Step 1: Find the GCF of the terms

In this case, the GCF of \(12x^2\) and \(18x\) is \(6x\), as both terms are divisible by \(6x\).

Step 2: Divide each term by the GCF

Now, divide both terms by the GCF (\(6x\)) and simplify: $$ \frac{12x^2}{6x} = 2x \\ \frac{18x}{6x} = 3 $$

Step 3: Write the factored polynomial

Finally, write the factored polynomial by combining the GCF (\(6x\)) and the simplified terms (\(2x\) and \(3\)): $$ 12x^2 + 18x = 6x(2x + 3) $$ So the factored polynomial is: $$ 6x(2x + 3) $$

Key Concepts

Greatest Common Factor (GCF)

When factoring polynomials, the greatest common factor (GCF) is your best friend. Think of the GCF as the biggest quantity that can be evenly divided from all the terms of an expression. Here are some helpful ways to identify the GCF:

• List all factors of each term.
• Find the largest factor common to all terms.

For example, for the terms \(12x^2\) and \(18x\), list their factors:

• \(12x^2 = 1, 2, 3, 4, 6, 12, x, x^2\)
• \(18x = 1, 2, 3, 6, 9, 18, x\)

The largest factor they share is \(6x\). By identifying the GCF, you can simplify polynomial expressions and make them easier to work with.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations. They do not have an equality sign like algebraic equations.Some key points to understand about algebraic expressions include:

• They consist of terms, such as \(12x^2\) and \(18x\) in the exercise.
• Each term is a product of constants (numbers) and variables (letters).
• Variables can have exponents to indicate repeated multiplication, as seen with \(x^2\).

Understanding how to manipulate these expressions is crucial. Factoring them, like in the exercise, is a way to break them down into simpler elements that are easier to handle and solve.

Polynomial Division

Polynomial division is a method used to simplify polynomials by dividing them by their factors. This process helps to break down complex expressions into more manageable parts.To divide polynomials effectively:

• Identify common factors in the polynomial terms. This is akin to finding the GCF.
• Divide each term by the common factor, as we did when dividing \(12x^2\) and \(18x\) by \(6x\).
• Simplify the resulting expression to its simplest form.

In our exercise, after dividing by the GCF of \(6x\), the expression simplifies to \(6x(2x + 3)\). This simplification makes the polynomial easier to work with, setting the foundation for further algebraic operations, like solving or expanding, if needed.