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Magazine Chapter 1: Problem 94
Factor the expression completely. $$x^{3}+3 x^{2}-x-3$$
Short Answer
Expert verified
The expression factors to \((x - 1)(x + 1)(x + 3)\).
Step by step solution
01
Group Terms
First, we look at the expression \(x^3 + 3x^2 - x - 3\) and group the terms to facilitate factoring by grouping. We rearrange it as \((x^3 + 3x^2) + (-x - 3)\).
02
Factor Out Common Factors
In the first group \(x^3 + 3x^2\), we can factor out \(x^2\), giving us \(x^2(x + 3)\). In the second group \(-x - 3\), we factor out \(-1\), giving us \(-1(x + 3)\).
03
Combine Like Terms
Observe that both factored groups contain a common binomial factor \((x + 3)\). Therefore, we can factor \((x + 3)\) out, resulting in \((x^2 - 1)(x + 3)\).
04
Factor the Difference of Squares
Recognize \(x^2 - 1\) as a difference of squares. Apply the formula \(a^2 - b^2 = (a - b)(a + b)\) with \(a = x\) and \(b = 1\). This gives \((x - 1)(x + 1)\).
05
Write the Complete Factorization
Substitute the factored expression from Step 4 back into the expression. Thus, the complete factorization of the original expression is \((x - 1)(x + 1)(x + 3)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring by Grouping
Factoring by grouping is a helpful technique used in algebra to factor polynomials, especially when they have four terms or more. This method involves grouping terms in a way that allows us to factor them easily. In this method, we first look for terms that can be paired together either through addition or subtraction.
- For example, in the expression \(x^3 + 3x^2 - x - 3\), we can group the terms into two different pairs: \((x^3 + 3x^2) + (-x - 3)\).
- Next, we factor out the greatest common factor from each group. For \(x^3 + 3x^2\), the common factor is \(x^2\), leaving us with \(x^2(x + 3)\).
- Similarly, in the group \(-x - 3\), we can factor out \(-1\) to get \(-1(x + 3)\).
Difference of Squares
The difference of squares is a special technique used to factor specific types of polynomial expressions. This method is useful because it simplifies expressions quickly by using a well-known identity. The difference of squares occurs when we have an expression of the form \(a^2 - b^2\). This expression can be rewritten as \((a - b)(a + b)\).
- In the context of the example problem, after employing factoring by grouping, we arrived at \(x^2 - 1\).
- Notice that \(x^2 - 1\) is a perfect example of a difference of squares since it can be seen as \(x^2 - 1^2\).
Polynomial Expressions
Polynomial expressions are central to algebra and are vital in various mathematical computations. They are formed by combining variables, coefficients, and exponents in a mathematical expression where the operations involved are only addition, subtraction, and multiplication.
- For example, the starting expression \(x^3 + 3x^2 - x - 3\) is a third-degree polynomial, as its highest exponent is three.
- Polynomials can be as simple as linear expressions, like \(x + 1\), or more complex with multiple terms, like \(x^3 + 3x^2 - x - 3\).
