Question

Factor \(6 x^{3}-2 x^{2}\).

Short answer

The factored form is \(2x^2(3x - 1)\).

Step-by-step solution

Find the Greatest Common Factor (GCF)

Identify the greatest common factor of the coefficients and the variable terms. For the expression \(6x^3 - 2x^2\), the GCF of the coefficients 6 and 2 is 2. The common factor in the variable terms \(x^3\) and \(x^2\) is \(x^2\). Therefore, the GCF is \(2x^2\).

Factor out the GCF

Divide each term in the expression \(6x^3 - 2x^2\) by the GCF \(2x^2\). This gives: \[\frac{6x^3}{2x^2} = 3x\] and \[\frac{2x^2}{2x^2} = 1\]. Now, factor out \(2x^2\) from the original expression: \(6x^3 - 2x^2 = 2x^2(3x - 1)\).

Key concepts

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is crucial when factoring polynomials.
The GCF is the largest number, including variables, that can divide each term without leaving a remainder.
For example, in the polynomial expression \(6x^3 - 2x^2\), it's essential to find the GCF of both the coefficients and the variable terms.
To find the GCF of the coefficients 6 and 2, you look at their greatest common divisor, which is 2.
Then, examine the variables \(x^3\) and \(x^2\). The common factor is \(x^2\). Therefore, the GCF is \(2x^2\).
Once you identify the GCF, you can factor it out to simplify the polynomial.

Polynomial Division

Polynomial division is the process of dividing each term in the polynomial by the GCF.
This step makes the polynomial easier to handle and understand.
Once we know the GCF \(2x^2\) for \(6x^3 - 2x^2\), we proceed by dividing each term:
\[ \frac{6x^3}{2x^2} = 3x \]
\[ \frac{2x^2}{2x^2} = 1 \]
By efficiently performing these divisions, we simplify the expression into its factored form.
This step helps break down the polynomial into smaller, more manageable terms.

Variable Terms

In polynomial expressions, variable terms denote the parts of the expression that contain variables like x, y, etc.
It’s vital to consider these variable terms when factoring polynomials.
Identifying the highest power of the variable common to each term is key in finding the GCF.
In the expression \(6x^3 - 2x^2\), both terms share the variable \(x\).
The lowest exponent of \(x\) common to both is \(x^2\).
This means \(x^2\) is part of the GCF, along with 2 (the GCD of the coefficients).
By factoring out \(2x^2\), we simplify the polynomial to:
\(6x^3 - 2x^2 = 2x^2(3x - 1)\).
Now, the polynomial is easier to work with.