Problem 87 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 3 x^{3}+3 x^{2}-27 x-27 $$

Step-by-Step Solution

Answer

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

Step 1: Identify Common Factors

First, identify any common factors in the entire polynomial. Notice that each term has a factor of 3, so factor out the 3: \[ 3(x^3 + x^2 - 9x - 9) \]

Step 2: Group Terms

After factoring the 3 out, proceed by grouping the terms inside the parentheses:\[ 3[(x^3 + x^2) + (-9x - 9)] \]

Step 3: Factor by Grouping

Take out the common factor from each group. The first group \((x^3 + x^2)\) has a common factor \(x^2\), and the second group \((-9x - 9)\) has a common factor \(-9\):\[ 3[x^2(x+1) - 9(x+1)] \]

Step 4: Factor the Common Binomial

Both terms now contain \((x+1)\) as a factor. Factor out \((x+1)\):\[ 3[(x+1)(x^2 - 9)] \]

Step 5: Factor the Difference of Squares

Notice that \(x^2 - 9\) is a difference of squares, which can be factored further:\[ x^2 - 9 = (x - 3)(x + 3) \]Substitute back:\[ 3[(x+1)(x-3)(x+3)] \]

Step 6: Review and Conclusion

Combine all factors to write the original polynomial in its fully factored form:\[ 3(x+1)(x-3)(x+3) \]

Key Concepts

Difference of Squares

The concept of "difference of squares" is a fundamental tool when factoring polynomials. To understand this, imagine two perfect squares subtracted from one another, such as \( a^2 - b^2 \). This expression can always be factored into the product of two binomials: \((a - b)(a + b)\).
This principle is incredibly useful when you need to simplify expressions quickly. For instance, in our original problem, you noticed \( x^2 - 9 \). This is a classic difference of squares, where \( a = x \) and \( b = 3 \).

• The expression \( x^2 - 9 \) is then factored as \( (x - 3)(x + 3) \).

Recognizing and using the difference of squares can make polynomial factoring much easier and faster, saving you a lot of hassle in longer problems.

Factor by Grouping

"Factor by grouping" is a useful method when dealing with polynomials of four or more terms. This approach involves grouping terms into smaller sections and then factoring each group individually.
In the polynomial \( x^3 + x^2 - 9x - 9 \), you can split the expression into two pairs:\

• First group: \( x^3 + x^2 \)
• Second group: \(-9x - 9\)

By focusing on one group at a time, look for common factors within each one. For the first group, \( x^2 \) is common, and for the second group, \(-9\) is common.

• First group factor: \( x^2(x + 1) \)
• Second group factor: \(-9(x + 1) \)

The reoccurrence of \((x + 1)\) indicates that it's a common factor for the entire expression. Pull out this common binomial to obtain: \[(x + 1)(x^2 - 9)\]. Factor by grouping simplifies the otherwise cumbersome expressions through active separation and factorization.

Common Factors

Identifying "common factors" is one of the first steps in simplifying polynomial expressions. It involves finding values or variables that are shared components of each term in the polynomial.
In the original expression \( 3x^3 + 3x^2 - 27x - 27 \), observe that each term contains a factor of 3. By factoring out the 3, the polynomial inside decreases in complexity:

• Original polynomial: \( 3(x^3 + x^2 - 9x - 9) \)
• Factored 3 out: \( (x^3 + x^2 - 9x - 9) \)

Factoring out common factors at the start makes it easier to apply other factoring methods later, such as factoring by grouping. It also reduces the degree of the polynomial within the parentheses, bringing it closer to its simplest form.

Polynomial Expressions

"Polynomial expressions" are algebraic expressions consisting of variables and coefficients, arranged in terms of varying powers. They can be as simple as a single term (monomial) or complex, with multiple terms (polynomial).
Understanding how to factor these expressions is crucial for simplifying algebraic problems. In the given exercise, the polynomial expression \( 3x^3 + 3x^2 - 27x - 27 \) had four terms. Each term contributes to the behavior of the expression.
Polynomials are often written in descending order by the power of their terms. Factoring a polynomial into its simplest components allows one to solve equations and understand their graphs more easily. This task requires a firm grasp of concepts like common factors, factoring by grouping, and the difference of squares. Mastery of these polynomial expression factoring methods is essential for academic and real-world applications of algebra.