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Chapter 10: Problem 468

Factor the greatest common factor from each polynomial. $$4 a^{3}-4 a b^{2}$$

Short Answer

Expert verified
4a(a^2 - b^2)

Step by step solution

01

Identify the Common Factors

First, identify the common factors in the terms of the polynomial. The given polynomial is: \[4a^3 - 4ab^2\] Both terms share the coefficients (4) and the variable 'a'.
02

Extract the Greatest Common Factor (GCF)

The GCF of the polynomial terms \[4a^3\] and \[4ab^2\] is \[4a\]. Extract the GCF from each term: \[4a^3 = 4a \times a^2\]\[4ab^2 = 4a \times b^2\]
03

Factor Out the GCF

Factor out \[4a\] from the polynomial: \[4a^3 - 4ab^2 = 4a(a^2 - b^2)\]

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Key Concepts

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

greatest common factor
In mathematics, the Greatest Common Factor (GCF) refers to the largest factor that two or more numbers or terms share. When factoring polynomials, identifying the GCF is often the first step.
The process involves:
  • Identifying the coefficients (numeric part) and the variables (alphabetic part).
  • Determining the common factors for both coefficients and variables.

For instance, in the polynomial \[4a^3 - 4ab^2\], both terms share a coefficient of 4 and a variable 'a'. Thus, the GCF is \[4a\]. Extracting the GCF can simplify the polynomial and make other factoring techniques more manageable.
polynomial terms
Polynomial terms are algebraic expressions that consist of a coefficient multiplied by variables raised to a power. Terms in a polynomial are separated by plus or minus signs.
  • A term like \[-4ab^2\] has a coefficient of -4, variables 'a' and 'b', and the exponent for 'b' is 2.
  • Understanding each part of a term is crucial for effective factoring.

For example, in the polynomial \[4a^3 - 4ab^2\], it's essential to see that \[4a^3\] involves 'a' to the power of 3 while \[4ab^2\] involves 'b' to the power of 2. Recognizing these differences helps in identifying common factors and applying appropriate factoring techniques.
factoring techniques
Factoring is a method of breaking down a polynomial into simpler terms or factors. Some essential techniques include:
  • Factoring out the GCF: This is usually the first step, as shown in the polynomial \[4a^3 - 4ab^2\]. By extracting \[4a\], we simplify it to \[4a(a^2 - b^2)\].
  • Group factoring: Applied when dealing with polynomials with more terms. Terms are grouped to factor common elements from each group.
  • Special products: Recognizing patterns such as the difference of squares can be helpful. For instance, \[a^2 - b^2\] can be factored into \[(a-b)(a+b)\].

Each technique has its context and can be used alone or combined with others to simplify a polynomial. Practice and experience help in choosing the most effective method for a given polynomial.

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