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Chapter 1: Problem 85

Factor the expression by grouping terms. $$2 x^{3}+x^{2}-6 x-3$$

Short Answer

Expert verified
The expression factors to \((2x + 1)(x^2 - 3)\).

Step by step solution

01

Group Terms

First, group the terms in pairs to make factoring easier. The expression is: \(2x^3 + x^2 - 6x - 3\). We can group them as \((2x^3 + x^2)\) and \((-6x - 3)\).
02

Factor Common Factors in Each Group

Next, factor out the greatest common factor from each group. In the first group, \((2x^3 + x^2)\), the common factor is \(x^2\). In the second group, \((-6x - 3)\), the common factor is \(-3\). So we get:\[ x^2(2x + 1) - 3(2x + 1) \]
03

Factor Out the Common Binomial

Notice that both grouped terms have a common binomial factor, \((2x + 1)\). Factor this binomial out:\[ (2x + 1)(x^2 - 3) \]
04

Verify the Factored Form

Finally, verify the factored form by expanding \((2x + 1)(x^2 - 3)\): \((2x + 1)(x^2 - 3) = 2x^3 - 6x + x^2 - 3 = 2x^3 + x^2 - 6x - 3\). The factored form is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Polynomial Expressions
Polynomial expressions are algebraic expressions that consist of variables and coefficients. These expressions are combined using operations such as addition, subtraction, and multiplication. A polynomial is composed of terms, and each term includes a coefficient (a numerical factor) and a variable raised to a certain power. Here's a quick rundown:
  • The highest power of the variable is called the degree of the polynomial.
  • A polynomial with one term is a monomial, two terms a binomial, and three terms a trinomial.
  • Our example, \(2x^3 + x^2 - 6x - 3\), is a polynomial with four terms.
Recognizing the structure of polynomials helps in factoring by grouping, as it allows you to see how to strategically pair terms for easier factoring.
Identifying the Greatest Common Factor
The greatest common factor (GCF) is the highest number or term that divides two or more numbers or terms without a remainder. When factoring an expression by grouping, identifying the GCF is a crucial step:
  • In the group \((2x^3 + x^2)\), the GCF is \(x^2\), because it is the highest power of \(x\) that divides both terms.
  • In the group \((-6x - 3)\), the GCF is \(-3\), since \(-3\) is the largest number that can divide both \(-6x\) and \(-3\).
Extracting the GCF from each group simplifies the expression, making it easier to spot common factors that can further be factored out.
Factoring Out Binomial Factors
Once you have factored out the greatest common factor from each term group, the next step is to identify binomial factors. A binomial is a polynomial with two terms. In our example, after factoring with GCF, we obtained:
  • \(x^2(2x + 1) - 3(2x + 1)\)
Here, \((2x + 1)\) is a common binomial factor in both terms resulting from the grouping. To factor this out, you write it once followed by a bracket containing the coefficients of the like terms:
  • \((2x + 1)(x^2 - 3)\)
Factoring by grouping often leads to identifying such binomial factors, making it easier to simplify complex polynomial expressions.

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