Question

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

Short answer

The factored form of the polynomial \(2 x^{3}-x^{2}-6 x+3\) is \((2x - 1)(x^2 - 3)\).

Step-by-step solution

Group the terms

Start by grouping the terms into two pairs. You choose the pairs that seem like they have common factors. Here, we can divide the polynomial as follows: \( (2x^3 - x^2) + (-6x + 3) \).

Factor out the GCF from each group

In the first group, the greatest common factor (GCF) is \(x^2\), and in the second group, it's \(-3\). Factoring these out, we get: \(x^2(2x - 1) - 3(2x - 1)\).

Factor out the common binomial term

Observe that the terms \(2x - 1\) are the same. You can factor this common binomial term out, to obtain: \((2x - 1)(x^2 - 3)\).

Key concepts

Polynomial Expressions

Polynomial expressions are mathematical phrases that include terms, which are expressions made up of variables and coefficients, combined using addition, subtraction, and sometimes multiplication. The general form of a polynomial expression looks like this: \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), where each \( a \) represents a coefficient, and \( n \) determines the degree of the polynomial or the highest power of the variable "\( x \)".
Polynomials can have various degrees:

• A constant polynomial with degree 0 has no variable component, for example, 5.
• A linear polynomial includes a variable, having a degree of 1, for example, \(2x + 3\).
• A quadratic polynomial has a degree of 2, such as \(x^2 + 5x + 6\).

In the exercise given \(2x^3 - x^2 - 6x + 3\), we see a cubic polynomial since the highest power of \(x\) is 3. This plays an essential role when it comes to the factorization process.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers. When factorizing a polynomial, finding the GCF is crucial as it helps simplify algebraic expressions into more manageable terms. For instance, consider the polynomial already split into groups:

• First group: \(2x^3 - x^2\)
• Second group: \(-6x + 3\)

In the first group, the greatest common factor is \(x^2\), because both terms are divisible by \(x^2\).
In the second group, the GCF is \(-3\), ensuring that when factored out, it results in positive terms inside parentheses, maintaining a structured expression.
By identifying and factoring out these GCFs, each polynomial group becomes simpler and helps reveal common factors in the next steps of factorization.

Factor by Grouping

The factor by grouping technique is an efficient method for simplifying polynomial expressions, especially when dealing with four-term polynomials. This method combines terms into two groups, identifies common factors in each, and then further simplifies the expression. Here’s a step-by-step guide based on the example polynomial: \(2x^3 - x^2 - 6x + 3\).
The first step is to arrange the polynomial into logical groupings of terms. In this case:

• Group 1: \(2x^3 - x^2\)
• Group 2: \(-6x + 3\)

Next, factor out the greatest common factor from each group. For the first group, factor out \(x^2\), resulting in \(x^2(2x - 1)\). For the second group, factor out \(-3\), giving \(-3(2x - 1)\). Notice both groups include the binomial \(2x - 1\).
Finally, the common binomial \(2x - 1\) is factored out from the expression, simplifying the polynomial to \((2x - 1)(x^2 - 3)\). This transformation greatly simplifies working with the original expression and can be crucial in solving equations involving polynomials.