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Chapter 6: Problem 38

Write in factored form by factoring out the greatest common factor. \(36 p^{3}-24 p\)

Short Answer

Expert verified
The factored form is \(12p(3p^2 - 2)\).

Step by step solution

01

- Identify the Greatest Common Factor (GCF)

First, find the greatest common factor of both terms in the expression. For the coefficients 36 and 24, the GCF is 12. For the variables, the GCF is the lowest power of p, which is p.
02

- Factor Out the GCF

Divide each term by the GCF to factor it out. This gives: \[ 36p^3 \text{ divided by } 12p = 3p^2 \] and \[ 24p \text{ divided by } 12p = 2. \] So, the expression becomes: \[ 12p(3p^2 - 2). \]

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Key Concepts

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

Greatest Common Factor (GCF)
When dealing with algebraic expressions, one essential concept is the Greatest Common Factor (GCF). The GCF is the largest number or expression that divides evenly into each term in an algebraic expression.
For example, in the expression \(36p^{3} - 24p\), we have two terms. We need to find both the numerical GCF and the variable GCF:
  • Numerical GCF: Look at the coefficients 36 and 24. The largest number that divides both 36 and 24 evenly is 12.
  • Variable GCF: We consider the variable part with the smallest power, in this case, the variable is \( p \), and the smallest power is \( p \) to the first power.
Hence, the GCF for the entire expression is \(12p\).
Factored Form
Factoring an algebraic expression involves rewriting it as a product of its factors. The factored form is crucial as it simplifies the expression and makes it easier to solve equations or evaluate functions. Let's factor our given expression using the GCF we identified.
Start with the expression \(36p^{3} - 24p\) and the GCF \(12p\).
  • Divide each term by \(12p\):
    • \( \frac{36p^3}{12p} = 3p^2 \)
    • \( \frac{24p}{12p} = 2 \)

    After dividing, the expression transforms to: \(12p(3p^2 - 2)\).
    This product of \(12p\) and \(3p^2 - 2\) is the factored form of the original expression. The expression is now elegantly simplified.
Algebraic Expressions
An algebraic expression is a combination of variables, numbers, and at least one algebraic operation (like addition, subtraction, multiplication, or division).
  • Examples include \(4x + 3\), \(7y^2 - 2y + 5\), and \(3a^3 - 14a + 56\).
In algebra, it's often useful to rewrite expressions in a form that makes them easier to work with, like in our factoring example.
When you break an expression down (or factor it), you're looking for commonalities that let you simplify the terms. By factoring out the GCF, you reduce the algebraic expression to a simpler form that is easier to handle in both calculations and solving equations. This process highlights the foundational algebra concepts making complex problems more tractable.

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