Problem 14 - College Algebra with Corequisite Support

Question

For the following exercises, solve the following polynomial equations by grouping and factoring. $$ 3 x^{3}-6 x^{2}-27 x+54=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 3, -3, 2\).

Step 1: Group the terms

Group the terms in pairs that can be factored together. The given expression is \(3x^3 - 6x^2 - 27x + 54\). Group these as \((3x^3 - 6x^2) + (-27x + 54)\).

Step 2: Factor out the common factors

Factor the greatest common factor from each group. For the first group, \(3x^3 - 6x^2\), the GCF is \(3x^2\), giving \(3x^2(x - 2)\). For the second group, \(-27x + 54\), the GCF is \(-27\), giving \(-27(x - 2)\).

Step 3: Factor by grouping

Since both groups contain \((x - 2)\), factor \((x - 2)\) out, getting \((3x^2 - 27)(x - 2)\).

Step 4: Simplify the expression

Further simplify \(3x^2 - 27\) by factoring out 3, resulting in \(3(x^2 - 9)\). Thus, the expression becomes \(3(x^2 - 9)(x - 2)\).

Step 5: Apply the difference of squares formula

Recognize \(x^2 - 9\) as a difference of squares: \(x^2 - 9 = (x - 3)(x + 3)\). Thus, the expression is \(3(x - 3)(x + 3)(x - 2)\).

Step 6: Solve for x

Set each factor equal to zero: 1.\(x - 3 = 0\) implies \(x = 3\)2.\(x + 3 = 0\) implies \(x = -3\)3.\(x - 2 = 0\) implies \(x = 2\).

Key Concepts

Factoring by Grouping

Factoring by grouping is an effective method to solve polynomial equations when the polynomial is not easily factorable by conventional means. This involves breaking down the polynomial into smaller, manageable groups that can be factored separately.

The fundamental idea is to rearrange and group terms in a way that each group shares a common factor. Let's consider the polynomial \(3x^3 - 6x^2 - 27x + 54 = 0\). Begin by grouping terms into pairs such as \((3x^3 - 6x^2)\) and \((-27x + 54)\).

• For the first group \((3x^3 - 6x^2)\), the greatest common factor (GCF) is \(3x^2\), leading to \(3x^2(x - 2)\).
• For the second group \((-27x + 54)\), the GCF is \(-27\), giving \(-27(x - 2)\).

Notice both groups include the factor \((x - 2)\). This enables you to factor the entire expression as \((3x^2 - 27)(x - 2)\). This way, complex polynomials can be reduced to simpler factors, making them easier to solve.

Difference of Squares

The difference of squares is a specific pattern in algebra that allows you to simplify expressions of the form \(a^2 - b^2\). This can be factored into \((a - b)(a + b)\). In our problem, this concept is applied after initial factoring.

Once the initial grouping and factoring steps reduce the expression to \(3(x^2 - 9)(x - 2)\), you identify \(x^2 - 9\) as a difference of squares. Here, \(a = x\) and \(b = 3\), thus \(x^2 - 9 = (x - 3)(x + 3)\).

• The expression \((x^2 - 9)\) can be rewritten as \((x - 3)(x + 3)\), further simplifying the original polynomial.
• This transformation is crucial in clearing complexities and getting the expression ready for solving.

Appreciating the pattern recognition in difference of squares is key to solving higher degree polynomials efficiently.

Solving Equations

Once the polynomial is fully factored, solving it involves setting each factor equal to zero. This method leverages the "Zero Product Property," which states that if a product of factors equals zero, at least one of the factors must be zero.

After simplifying the polynomial \(3(x - 3)(x + 3)(x - 2)\), each factor can be individually set to zero:

• For \(x - 3 = 0\), solve to find \(x = 3\).
• For \(x + 3 = 0\), solve to find \(x = -3\).
• For \(x - 2 = 0\), solve to find \(x = 2\).

The solutions to the equation are \(x = 3, -3,\) and \(2\). These values satisfy the original polynomial equation, providing the necessary roots or solutions. Solving equations using factored expressions simplifies the process immensely.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that divides each term of the polynomial evenly. Identifying the GCF is an essential step in simplifying polynomial expressions, paving the way for further factoring techniques.

In the given polynomial \(3x^3 - 6x^2 - 27x + 54\), finding the GCF in smaller grouped parts enables simplification. For instance:

• In \(3x^3 - 6x^2\), the GCF is \(3x^2\), reducing it to \(3x^2(x - 2)\).
• In \(-27x + 54\), the GCF is \(-27\), making it \(-27(x - 2)\).

Using the GCF helps in rearranging and breaking down terms efficiently. This initial step not only aids in further factoring by grouping or other methods but also ensures each term is simplified to its most basic form. Understanding how to identify and extract the GCF is key to tackling polynomial equations effectively.