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Chapter 5: Problem 129

Factor each polynomial completely. \(4 x^{2}+4 x+1-y^{2}\)

Short Answer

Expert verified
\[ (2x + 1 + y)(2x + 1 - y) \]

Step by step solution

01

- Recognize the Difference of Squares

Notice that the polynomial can be written in a form that reveals a difference of squares. The expression can be viewed as \[ (2x + 1)^2 - y^2 \]
02

- Apply the Difference of Squares Formula

The difference of squares formula is \[ a^2 - b^2 = (a + b)(a - b) \]Here, let \[ a = (2x + 1) \] and \[ b = y \]. Thus, \[ (2x + 1)^2 - y^2 = ((2x + 1) + y)((2x + 1) - y) \]
03

- Write the Final Factored Form

After applying the formula, the expression factors completely to:\[ (2x + 1 + y)(2x + 1 - y) \]

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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 key concept when working with polynomial factorization. It occurs when you have two squared terms subtracted from one another. The general form is given as \[ a^2 - b^2 \] where \(a\) and \(b\) represent any expressions.
Here, the expression \((2x + 1)^2 - y^2\) fits perfectly into this format.
Notice how \((2x + 1)\) replaces \(a\) and \(y\) represents \(b\). Understanding this makes it easier to apply the difference of squares formula.
Polynomial Factorization
Polynomial factorization involves breaking down a polynomial into simpler terms called 'factors' which, when multiplied together, give back the original polynomial. For the given exercise, the polynomial \(4x^2 + 4x + 1 - y^2\) was first rewritten as \((2x + 1)^2 - y^2\).
This step was crucial because it reveals the polynomial as a difference of squares. Identifying patterns and rewriting expressions in simpler forms helps in successfully factoring polynomials. Factorization simplifies complex problems and makes solving equations much easier.
Factoring Techniques
Different techniques can be used to factor polynomials. Here are some useful techniques:
  • **Key Formula**: The difference of squares formula, \(a^2 - b^2 = (a + b)(a - b)\), is among the most straightforward methods for factorization.
  • **Trial and Error**: Sometimes factors are not easily recognizable. A trial and error approach, checking for possible factors, can be used.
  • **Grouping**: If a polynomial can be grouped into sets that are easier to factor separately, this technique can be useful.
For our problem, the key technique was recognizing and using the difference of squares formula. After rewriting \((2x + 1)^2 - y^2\) we identified \(a = 2x + 1\) and \(b = y\). Applying the formula, it was factored to its simplest form as: \((2x + 1 + y)(2x + 1 - y)\).
Each of these techniques, when mastered, can make polynomial factorization approachable and manageable.

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