Problem 70 - Algebra and Trigonometry

Question

Factor the expression completely. $$ x^{3}+3 x^{2}-x-3 $$

Step-by-Step Solution

Answer

The completely factored form of the expression is \((x + 3)(x - 1)(x + 1)\).

Step 1: Group Terms in Pairs

To simplify factorization, group terms into pairs: \[ (x^3 + 3x^2) + (-x - 3) \]

Step 2: Factor Out the Greatest Common Factor from Each Pair

Factor out the greatest common factor (GCF) from each pair of terms. From the first pair \((x^3 + 3x^2)\), factor out \(x^2\), and from the second pair \((-x - 3)\), factor out \(-1\): \[ x^2(x + 3) - 1(x + 3) \]

Step 3: Factor by Grouping

Notice the common factor \((x + 3)\) in both terms. Factor \((x + 3)\) out of the entire expression:\[ (x + 3)(x^2 - 1) \]

Step 4: Factor the Difference of Squares

The expression \(x^2 - 1\) is a difference of squares, which can be factored as \((x - 1)(x + 1)\):\[ (x + 3)(x - 1)(x + 1) \]

Step 5: Confirmation of Complete Factorization

Verify that the expression \((x + 3)(x - 1)(x + 1)\) is completely factored by expanding to ensure it matches the original expression \(x^3 + 3x^2 - x - 3\). The factorization is confirmed to be correct.

Key Concepts

Difference of Squares

The difference of squares is a special type of polynomial that can easily be factored. It follows the format

• n^2 - m^2

which can be factored into

• (n - m)(n + m)

. This process utilizes the fact that the product of conjugate pairs results in a

• difference of squares.

In the context of the given problem,

• \(x^2 -1\)

is a classic example of this. By recognizing that it can be rewritten as

• \((x - 1)(x + 1)\)

, we are able to simplify it, making it one of the last crucial steps in our factorization efforts. Understanding this concept will help you tackle many polynomial problems where parts of the expression fit the difference of squares pattern.

Common Factor

Finding and factoring out the greatest common factor (GCF) is often the first step in polynomial factorization. The GCF is the largest factor that divides each term in a polynomial. By identifying it, you can simplify expressions considerably and make them easier to work with. In the exercise, we first examine a pair of terms:

• \((x^3 + 3x^2)\)

From this pair, \(x^2\) is the greatest factor that can be evenly divided from both terms. When we factor \(x^2\) out, the expression becomes

• \(x^2(x + 3)\)

Similarly, for the pair

• \((-x - 3)\)

, \(-1\) is the GCF, leading to

• \(-1(x + 3)\)

. Mastering factoring out the GCF is a critical skill in tackling complex algebraic expressions.

Factor by Grouping

Factoring by grouping is an invaluable technique when dealing with four-term polynomials. It involves rearranging and grouping terms in a way that allows for simple factoring. In this exercise, we first rearrange the polynomial into pairs:

• \((x^3 + 3x^2) + (-x - 3)\)

. From each pair, we factor out the GCF, which results in

• \(x^2(x + 3) - 1(x + 3)\)

. Notice here, that

• \((x + 3)\)

is a common factor in both terms, allowing us to group them as:

• \((x + 3)(x^2 - 1)\)

. The technique of grouping is particularly useful when individual pairs in a polynomial share common factors, revealing a simple factorization with few steps.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. Understanding how to manipulate these forms is the groundwork of algebra. The given problem,

• \(x^{3}+3x^{2}-x-3\)

represents a typical polynomial expression that requires factorization. By applying basic algebraic principles like factoring by grouping and recognizing patterns such as differences of squares, we simplify complex expressions and solve equations more easily. Knowing how to rewrite and transform algebraic expressions opens the door to solving larger parts of mathematics, including quadratic functions and beyond.