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Chapter 5: Problem 48

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

Short Answer

Expert verified
\(9 a^{4} - a^{2} b^{2} = a^{2}(3a + b)(3a - b)\)

Step by step solution

01

- Identify the common factor

Examine the terms in the expression. Notice that both terms have a common factor of \(a^{2}\).
02

- Factor out the common factor

Factor out \(a^{2}\) from both terms: \[9 a^{4} - a^{2} b^{2} = a^{2}(9 a^{2} - b^{2})\]
03

- Recognize the difference of squares

Observe that \(9 a^{2} - b^{2}\) is a difference of squares, which can be factored further using the identity \(x^{2} - y^{2} = (x + y)(x - y)\).
04

- Apply the difference of squares

Apply the difference of squares to factor \(9 a^{2} - b^{2}\): \[9 a^{2} - b^{2} = (3a + b)(3a - b)\]
05

- Combine the factors

Combine the results into the fully factored form: \[9 a^{4} - a^{2} b^{2} = a^{2}(3a + b)(3a - b)\]

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Key Concepts

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

Common Factor
The first step in factoring any polynomial is to look for a common factor. A common factor is a term that is present in each term of the polynomial. In the exercise, the polynomial is \(9a^{4} - a^{2}b^{2}\). Both terms in this polynomial contain the factor \(a^{2}\). Factoring out the common factor involves dividing each term by \(a^{2}\) and pulling it out to the front. This results in:
  • \(9a^{4}/a^{2} = 9a^{2}\)
  • \(-a^{2}b^{2}/a^{2} = -b^{2}\)
Combining these gives us: \(a^{2}(9a^{2} - b^{2})\)
Factoring out common factors simplifies the polynomial and makes further steps easier.
Difference of Squares
The next concept used in the solution is the difference of squares. This is a specific algebraic pattern where you have two terms squared and separated by a subtraction sign. The general form is:\[x^{2} - y^{2} = (x + y)(x - y)\]In this exercise, after factoring out the common factor, we are left with \(9a^{2} - b^{2}\). This can be identified as a difference of squares because:
  • \(9a^{2}\) is a perfect square \((3a)^{2}\)
  • \(-b^{2}\) is a perfect square \((-b)^{2}\)
Applying the difference of squares formula, we get:\[9a^{2} - b^{2} = (3a + b)(3a - b)\]
This method is valuable because it breaks down complex expressions into simpler binomials.
Polynomial Factorization
Polynomial factorization is the process of expressing a polynomial as a product of its factors. In this exercise, we combined multiple factorization techniques. Start by factoring out the common factor, \(a^{2}\), then recognize and apply the difference of squares pattern.Polynomial factorization often involves several steps:
  • First, identify and factor out any common factors.
  • Next, look for special products like the difference of squares, perfect square trinomials, or the sum/difference of cubes.
  • Finally, combine all your factors to express the polynomial in its simplest factored form.
For our example:\[9a^{4} - a^{2}b^{2} = a^{2}(3a + b)(3a - b)\]
By breaking down the polynomial into smaller and easier-to-handle factors, we simplify solving equations or finding polynomial roots.

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