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Magazine Chapter 1: Problem 25
Factor the polynomial. $$3 x^{3}+3 x^{2}-27 x-27$$
Short Answer
Expert verified
The polynomial factors to \(3(x + 1)(x + 3)(x - 3)\).
Step by step solution
01
Identify Common Factors
First, examine if there is a common factor in each term of the polynomial. Here, every term in the polynomial has a common factor of 3. Factoring out 3 gives: \[ 3(x^3 + x^2 - 9x - 9) \].
02
Group Terms
Next, look at the expression inside the parentheses: \( x^3 + x^2 - 9x - 9 \). Separate the terms into two groups where each group can potentially be factored further: \( (x^3 + x^2) + (-9x - 9) \).
03
Factor by Grouping
Now, factor out the greatest common factor from each group: For the first group \( x^3 + x^2 \), factor out \( x^2 \) to get \( x^2(x + 1) \).For the second group \( -9x - 9 \), factor out \( -9 \) to get \( -9(x + 1) \).Thus, the expression becomes: \[ x^2(x + 1) - 9(x + 1) \].
04
Factor Out Common Binomial
Both terms have a common binomial \( x + 1 \). Factor this out to get: \[ (x + 1)(x^2 - 9) \].
05
Factor Remaining Quadratic
Notice that \( x^2 - 9 \) is a difference of squares, which can be factored further into: \[ (x + 3)(x - 3) \]. So, the expression becomes: \[ (x + 1)(x + 3)(x - 3) \].
06
Combine All Factors
Finally, include the factor of 3 factored out in the first step along with the complete factorization inside the parentheses. The fully factored expression is: \[ 3(x + 1)(x + 3)(x - 3) \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Common Factor
Identifying the common factor in a polynomial is the doorway to simplifying the expression. Essentially, a common factor is a number or variable that divides each term of the polynomial without a remainder. In the given polynomial:
\[3x^3 + 3x^2 - 27x - 27\],
we start by checking each term. Here, all terms
\[3(x^3 + x^2 - 9x - 9)\].This simplifies the work needed for further factoring, setting a streamlined path ahead.
\[3x^3 + 3x^2 - 27x - 27\],
we start by checking each term. Here, all terms
- \(3x^3\)
- \(3x^2\)
- \(-27x\)
- \(-27\)
\[3(x^3 + x^2 - 9x - 9)\].This simplifies the work needed for further factoring, setting a streamlined path ahead.
Grouping Method
The grouping method is a clever tactic for factoring polynomials, especially when dealing with four-term polynomials like
\(x^3 + x^2 - 9x - 9\).
The idea is to separate the polynomial into two groups. You organize it into pairs, aiming to factor out a common factor in each group. For the given polynomial:1. Group the terms as \((x^3 + x^2)\) and \((-9x - 9)\).2. For each group, extract the greatest common factor: - In the first group, \(x^2\) is common, so we factor it out, leading to \(x^2(x+1)\). - The second group shares \(-9\) as a factor, giving us \(-9(x+1)\).
After factoring each group, the expression reads as \(x^2(x+1) - 9(x+1)\).Since both groups now have the \(x+1\) factor, this common binomial is factored out: \((x+1)(x^2 - 9)\). It鈥檚 like solving a puzzle鈥攆inding one piece that fits perfectly into both sides, revealing a clearer picture.
\(x^3 + x^2 - 9x - 9\).
The idea is to separate the polynomial into two groups. You organize it into pairs, aiming to factor out a common factor in each group. For the given polynomial:1. Group the terms as \((x^3 + x^2)\) and \((-9x - 9)\).2. For each group, extract the greatest common factor: - In the first group, \(x^2\) is common, so we factor it out, leading to \(x^2(x+1)\). - The second group shares \(-9\) as a factor, giving us \(-9(x+1)\).
After factoring each group, the expression reads as \(x^2(x+1) - 9(x+1)\).Since both groups now have the \(x+1\) factor, this common binomial is factored out: \((x+1)(x^2 - 9)\). It鈥檚 like solving a puzzle鈥攆inding one piece that fits perfectly into both sides, revealing a clearer picture.
Difference of Squares
One of the special cases in polynomial factoring is the difference of squares. The expression takes the form \(a^2 - b^2\), and it factors into two binomials as \((a + b)(a - b)\). For instance, in the previous step, we arrived at \(x^2 - 9\) as one of the factors.
Recognizing this as a difference of squares is key:
\[(x+3)(x-3)\].
Thus, combining this factorization with the previous one gives \((x+1)(x+3)(x-3)\).
Understanding and recognizing patterns like this eases the factorization process considerably, allowing one to break down complex problems into simple, recognizable forms.
Recognizing this as a difference of squares is key:
- The term \(x^2\) is \((x)^2\), and
- the term \(9\) is \((3)^2\).
\[(x+3)(x-3)\].
Thus, combining this factorization with the previous one gives \((x+1)(x+3)(x-3)\).
Understanding and recognizing patterns like this eases the factorization process considerably, allowing one to break down complex problems into simple, recognizable forms.
