Problem 63 - Elementary Algebra

Question

List, if any should appear, the common factors in the expression \(3 x^{4}+6 x^{3}-18 x^{2}\).

Step-by-Step Solution

Answer

Answer: The common factors in the expression are 3 and \(x^2\).

Step 1: Identify the common factors of the coefficients

The coefficients of the given expression are 3, 6, and -18. The common factor among these coefficients is 3.

Step 2: Identify the common factors of the variable powers

The variable powers in the expression are \(x^4\), \(x^3\), and \(x^2\). The common variable power among these three terms is \(x^2\) since it's the lowest power of x.

Step 3: Factor out the common factors

Now that we have identified the common factors of the coefficients (3) and the common variable power (\(x^2\)), we can factor these out from our expression. Factoring out 3 and \(x^2\), we get: \(3x^2( x^2 + 2x - 6)\)

Step 4: Simplify the expression (if possible)

We can check if the expression inside the parentheses can be simplified further. In this case, the expression \(x^2 + 2x - 6\) cannot be factored further. So, the final factored form of the given expression is: \(3x^2( x^2 + 2x - 6)\) The common factors in the expression are 3 and \(x^2\).

Key Concepts

Common Factors

A common factor in an algebraic expression is a term that evenly divides all other terms in that expression. Identifying common factors is a powerful tool for simplifying and solving algebraic equations. When you observe the expression 3x^4 + 6x^3 - 18x^2, the first step is to look at the coefficients: 3, 6, and -18. These numbers all share 3 as a factor. Having a keen eye for such details is crucial for simplifying algebraic expressions.

Recognizing common factors can sometimes feel like a puzzle. Imagine having a set of building blocks where only certain sizes can stack together perfectly without any leftovers. In this context, if every term in the expression can be divided by that 'size' or factor, then you've found a common factor that can be pulled out, simplifying the rest of the expression.

It's also essential to communicate to students that common factors could include numerical values, like our example, or variables, which we will discuss in the next section.

Variable Powers

Understanding variable powers is crucial when dealing with algebraic expressions. A variable raised to a power, like x^2 or x^3, indicates repeated multiplication of the variable. When you see an expression like 3x^4 + 6x^3 - 18x^2, it's like looking at towers of blocks, with each x being a block and the exponent telling you how many blocks are in the tower.

When factoring, you want to find the shortest tower, or the lowest power of x, that is still present in all the other towers. This is the greatest common factor in terms of the variable power. Here, the shortest tower among x^4, x^3, and x^2 is x^2. By selecting it, you can 'take down' that many blocks from each tower, making the rest of the problem more manageable.

Simplifying Expressions

Simplifying an expression involves reducing it to its most basic form without changing its value. It's like cleaning up a messy room so you can see everything clearly. For algebraic expressions, simplifying can mean combining like terms, factoring, or canceling out inverses. Students should think of simplifying as organizing their mathematical 'room.'

In our problem, once we factored out the common numerical and variable factors, we were left with x^2 + 2x - 6 inside the parentheses. This expression does not have any like terms to combine and cannot be factored further, indicating that we have simplified it as much as possible. Simplifying expressions is a key step in achieving an easier-to-understand problem, leading to a clearer path toward the solution.

Algebraic Factorization

Algebraic factorization is the reverse of expanding and involves breaking down algebraic expressions into simpler, multiplied factors, much like deconstructing a complex structure into its building blocks. In the context of our example, factorization translates to pulling out the common factors of 3x^2 from each term of the expression 3x^4 + 6x^3 - 18x^2. What remains is an expression that looks less complex and is more understandable.

Factorization allows students to tackle complicated expressions step by step, examining each piece until the expression is fully simplified. By mastering algebraic factorization, students gain confidence in rearranging and solving expressions in a structured manner, enabling them to approach more complex algebra problems with ease.