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Chapter 5: Problem 51

Factor completely. \(9 a^{4}-25 a^{2} b^{4}\)

Short Answer

Expert verified
The factored form is \[ (3a^2 - 5ab^2)(3a^2 + 5ab^2) \].

Step by step solution

01

- Identify the common factor

Notice that the given expression is a difference of two squares. Rewrite the expression as the difference of two squares: \[ 9a^4 - 25a^2b^4 = (3a^2)^2 - (5ab^2)^2 \]
02

- Apply the difference of squares formula

Use the identity for the difference of squares, which is \[ A^2 - B^2 = (A - B)(A + B) \]. Here, let \[ A = 3a^2 \] and \[ B = 5ab^2 \]. Thus, \[ (3a^2 - 5ab^2)(3a^2 + 5ab^2) \]
03

- Factor each resulting binomial if possible

In this case, neither \[3a^2 - 5ab^2\] nor \[3a^2 + 5ab^2\] can be factored further, because they are both already in their simplest factored form.
04

Final Expression

Thus, the completely factored form of the given expression is: \[ (3a^2 - 5ab^2)(3a^2 + 5ab^2) \]

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Key Concepts

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

Difference of Squares
The 'difference of squares' is a crucial algebraic concept. It applies when we have two squared terms separated by a minus sign. This pattern can be factored using the formula: \[ A^2 - B^2 = (A - B)(A + B) \] Understanding this formula is essential because it simplifies complex expressions. In the given exercise, we noticed a difference of squares in \[ 9a^4 - 25a^2b^4 \] By rewriting each term as a square鈥攍ike \[ (3a^2)^2 \] and \[ (5ab^2)^2 \]鈥攚e can apply the formula easily.
Binomials
A binomial is an algebraic expression containing two distinct terms. For example, in our problem, \[ 3a^2 - 5ab^2 \] and \[ 3a^2 + 5ab^2 \] are both binomials. These are crucial in factoring because they often appear in simplified forms after applying various formulas or identities. When factoring polynomials, recognizing binomials helps us use specific rules effectively, like the difference of squares rule we used in the solution.
Factoring Algebraic Expressions
Factoring algebraic expressions involves breaking down a complex expression into simpler components. The goal is to make solving equations easier. In the example, we first rewrote the expression \[ 9a^4 - 25a^2b^4 \] as a difference of squares, then factored it into binomials. Here are steps to factor any algebraic expression:
  • Identify if there is a common factor.
  • Check if the expression is a difference of squares, trinomial, or another pattern.
  • Apply the appropriate factoring formula.
  • Simplify the factors if possible.
Practice these steps to master factoring algebraic expressions!

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