Problem 25 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The polynomial factors as \(3(x - 3)(x + 3)(x + 1)\).

Step 1: Identify the Common Factor

Look at the polynomial \(3x^3 + 3x^2 - 27x - 27\) and notice that each term has a factor of 3. Factor out the 3 from the entire polynomial to simplify it: \[3(x^3 + x^2 - 9x - 9)\]

Step 2: Group Terms for Simplification

Now, consider the expression inside the parenthesis: \(x^3 + x^2 - 9x - 9\). Group the terms in pairs: \[(x^3 + x^2) + (-9x - 9)\]

Step 3: Factor Each Group

Factor out the greatest common factor from each pair of terms. For \((x^3 + x^2)\), the common factor is \(x^2\):\[x^2(x + 1)\]For \((-9x - 9)\), the common factor is \(-9\):\[-9(x + 1)\]

Step 4: Factor by Grouping

Notice that \((x + 1)\) is a common factor in both groups. Factor \((x + 1)\) out of the entire expression:\[x^2(x + 1) - 9(x + 1) = (x^2 - 9)(x + 1)\]

Step 5: Factor the Difference of Squares

In the expression \(x^2 - 9\), recognize this as a difference of squares:\[x^2 - 9 = (x - 3)(x + 3)\]So, the polynomial becomes:\[3(x - 3)(x + 3)(x + 1)\]

Key Concepts

Common Factor

Finding a common factor in a polynomial means identifying a number or expression that divides each term completely without leaving a remainder. This step is crucial because it simplifies the polynomial, making it easier to factor further.

In the given polynomial, \(3x^3 + 3x^2 - 27x - 27\), each term has a factor of 3. By factoring out this common factor, we simplify the polynomial to \(3(x^3 + x^2 - 9x - 9)\).

This reduction helps in the subsequent steps of factoring. Removing the common factor makes it easier to recognize patterns or further factorize the expression.

Grouping Method

The grouping method is a technique used to factor polynomials with four or more terms by organizing terms into pairs or groups. This approach helps in spotting common factors within these groups.

In the example, \(x^3 + x^2 - 9x - 9\), we group the terms as \((x^3 + x^2) + (-9x - 9)\).

Inside each group, we can factor out a common factor. From \((x^3 + x^2)\), we extract \(x^2\), leaving us with \(x^2(x + 1)\).
Similarly, \((-9x - 9)\) has \(-9\) as a common factor, resulting in \(-9(x + 1)\).

This method reveals a shared factor of \((x + 1)\) across both groups, allowing us to factor the entire expression further as \((x^2 - 9)(x + 1)\).

Difference of Squares

The difference of squares is a special pattern where a polynomial takes the form \(a^2 - b^2\). This can be easily factored into \((a - b)(a + b)\). Recognizing this pattern is important because it simplifies the factoring process.

For the expression \(x^2 - 9\), we recognize it as a difference of squares since \(9\) is \(3^2\). It can be rewritten as \(x^2 - 3^2\), which factors into \((x - 3)(x + 3)\).

This step completes the factorization of the polynomial, rendering it as \((x - 3)(x + 3)(x + 1)\), still multiplied by the common factor 3 extracted earlier. Understanding this concept helps students factor similar expressions quickly.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Polynomials can have significant applications in both algebra and calculus, varying from simple to complex forms.

In the problem demonstrated, the function was initially \(3x^3 + 3x^2 - 27x - 27\). Factoring allows us to express this polynomial in a product of simpler terms. This provides insight into the roots of the polynomial and often simplifies solving equations or graphing functions.

Factoring polynomials is not only a key skill in algebra but also aids in understanding complex polynomial behavior, their zeroes, and their turning points. This makes polynomial functions a fundamental part of higher mathematical studies and applications.