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Chapter 5: Problem 46

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

Short Answer

Expert verified
The complete factorization is (x^{2}(3x - 5)(3x + 5)).

Step by step solution

01

Identify Common Factor

The given expression is (9x^{4} - 25x^{2}). Start by identifying any common factors in the terms. In this case, both terms share a common factor of \(x^{2}\).
02

Factor Out the Common Factor

Factor \(x^{2}\) out from both terms. This gives: (9x^{4} - 25x^{2}) = x^{2}(9x^{2} - 25).
03

Recognize and Factor the Difference of Squares

The expression inside the parentheses, (9x^{2} - 25), is a difference of squares. Recall that (a^{2} - b^{2}) = (a - b)(a + b). Here, (9x^{2}) is (3x)^{2} and (25) is (5)^{2}.
04

Apply the Difference of Squares Formula

Applying the difference of squares formula, factor (9x^{2} - 25) into: (3x - 5)(3x + 5)
05

Write the Complete Factorization

Finally, include the common factor x^{2} back into the expression: (9x^{4} - 25x^{2}) = x^{2}(3x - 5)(3x + 5)

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Key Concepts

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

Common Factor
Factoring polynomials often starts with identifying the common factors. These are the numbers, variables, or terms that are present in each term of the polynomial.

In our example, the expression is: \(9x^{4} - 25x^{2}\).

This means we check both terms for any factors they share. Here, both terms contain \(x^{2}\). By factoring out \(x^{2}\), the expression simplifies to:
\[ x^{2}(9x^{2} - 25)\]

This initial step often makes the remaining problem easier to solve. By removing the common factor first, we reduce the complexity of the polynomial.
Difference of Squares
The difference of squares is a special pattern that simplifies the factorization of certain expressions. It applies to any expression in the form \(a^{2} - b^{2}\). Remember, this pattern states:

\[ (a^{2} - b^{2}) = (a - b)(a + b) \]

For our example \(9x^{2} - 25\), we can see that it fits the difference of squares pattern:
\[ 9x^{2} = (3x)^{2} \]
and
\[ 25 = 5^{2} \]

Thus, we can apply the formula to factor \(9x^{2} - 25\). This gives us:
\[ (3x - 5)(3x + 5) \]
Recognizing the difference of squares saves time and simplifies the factoring process significantly.
Factorization
Factorization is the process of breaking down a complicated polynomial into simpler components or factors. In our solved exercise, the original polynomial was \(9x^{4} - 25x^{2}\).

Our steps were as follows:
  • Identify the common factor \(x^{2}\).
  • Factor out \(x^{2}\) to simplify the polynomial to \(x^{2}(9x^{2} - 25)\).
  • Use the difference of squares formula to factor \(9x^{2} - 25\).
  • Combine all the factors to get the final factorized form: \(x^{2}(3x - 5)(3x + 5)\).


This systematic approach ensures that every polynomial is broken down as much as possible, making them easier to work with in algebraic problems.

Understanding these methods is crucial for mastering polynomial factorization.

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