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Chapter 4: Problem 12

Determine the missing factor. \(\quad 6 x^{5}-9 x^{3}-3 x=3 x(\quad ?)\)

Short Answer

Expert verified
The missing factor is \(2x^4 - 3x^2 - 1\).

Step by step solution

01

Identify Common Factors

First, identify the common factors in each term on the left-hand side of the equation. The terms are \(6x^5\), \(-9x^3\), and \(-3x\). Notice that each term contains a factor of \(x\) and \(3\) can be factored out as it divides each of the coefficients (6, 9, and 3).
02

Factor Out the Common Term

Factor the common term \(3x\) out from the left-hand side of the equation:\[6x^5 - 9x^3 - 3x = 3x(2x^4 - 3x^2 - 1)\]This step shows that when you multiply \(3x\) by \(2x^4 - 3x^2 - 1\), you get back the original expression.
03

Write the Expression in Factored Form

Now that you have factored out \(3x\), rewrite the equation showing the factored portion:\[6x^5 - 9x^3 - 3x = 3x(2x^4 - 3x^2 - 1)\]This reveals that the unknown factor is \(2x^4 - 3x^2 - 1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Factor
Identifying a common factor is the first and essential step when we approach polynomial factoring. A **common factor** is a number or an algebraic expression that divides each term of the polynomial equally without leaving a remainder. For the expression given: \( 6x^5 - 9x^3 - 3x \), each term shares the factors of \(x\) and \(3\). This means:
  • Every term contains the variable \(x\), making \(x\) a part of the common factor.
  • The coefficients 6, 9, and 3 are all divisible by 3. So, 3 is another common factor for the numerical part.
By recognizing these factors, we can simply "pull out" \(3x\) from each term in the polynomial.
Factoring the common factor is not only a crucial skill in algebra but also helps simplify expressions, making them easier to work with in equations or further operations.
Factoring Process
The factoring process involves breaking down a complex algebraic expression into simpler components, called factors. These factors, when multiplied, will yield the original expression. Here's how the process applies to our example:
Initially, we identified the common factor \(3x\). The next step is extracting the \(3x\) out from each term in \(6x^5 - 9x^3 - 3x\). This procedure results in:
  • Divide each term \(6x^5\), \(-9x^3\), and \(-3x\) by \(3x\).
  • After factoring \(3x\) out, the resulting expression is written as \(3x(2x^4 - 3x^2 - 1)\).
The beauty of this process lies in its ability to simplify polynomial equations or expressions, allowing for easier manipulation鈥攑articularly useful for solving equations or finding roots.
Algebraic Expression
An algebraic expression is a mathematical phrase combining numbers, operations, and variables. In our example, the original expression is \(6x^5 - 9x^3 - 3x\). A crucial part of working with algebraic expressions is understanding their structure and simplifying them through techniques like factoring.
Here鈥檚 a breakdown:
  • Algebraic expressions like \(6x^5 - 9x^3 - 3x\) often result from adding or subtracting monomials.
  • Simplifying them often involves recognizing common factors or employing other algebraic techniques for rearrangement, such as collecting like terms or using identities.
Understanding and effectively manipulating algebraic expressions is a cornerstone of algebra. It enables students to perform a wide range of mathematical tasks, from solving equations to modeling real-world scenarios.

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