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Chapter 5: Problem 70

Factor. $$4 x^{2}+4 x+1-y^{2}$$

Short Answer

Expert verified
The factorized form of the given expression is \((2x+1-y)(2x+1+y)\).

Step by step solution

01

Identify the Quadratic Equation

The given expression is \(4 x^{2}+4 x+1-y^{2}\). From this expression we can see that, the quadratic part \((4 x^{2}+4 x+1)\) is a perfect square trinomial, \((2x+1)^{2}\).
02

Re-write the Expression

Now, re-write the expression, replacing the quadratic part we identified as a perfect square trinomial, we get the expression to be \((2x+1)^{2}-y^{2}\)
03

Use of Difference of Squares Formula

Next, we use the 'difference of squares' formula, \(a^{2}-b^{2}=(a-b)(a+b)\). Using this formula the expression becomes \((2x+1-y)(2x+1+y)\).
04

Expression Simplification

Now, simplify the expression further if possible. Here, it cannot be further simplified.

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Key Concepts

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

Perfect Square Trinomial
A perfect square trinomial is an important part of algebraic factoring that makes expressions much easier to handle. It is an expression that can be written in the form
  • \(a^2 + 2ab + b^2 = (a+b)^2\)
  • or \(a^2 - 2ab + b^2 = (a-b)^2\)
Perfect square trinomials appear frequently when expanding binomials that are squared. For example, consider \(4x^2 + 4x + 1\). Here, if you examine closely,
  • the first term, \(4x^2\), is a square of \(2x\).
  • The last term, \(1\), is a square of \(1\).
  • The middle term, \(4x\), satisfies the condition \(2ab\) as \(2(2x)(1) = 4x\).
Thus, this trinomial can be factored as \((2x+1)^2\). It is a handy shortcut that saves you plenty of time when working with quadratic expressions.
Difference of Squares
The difference of squares is a nifty algebraic identity and is characterized by an expression where two squares are subtracted from each other. This formula is expressed as
  • \(a^2 - b^2 = (a-b)(a+b)\)
It is crucial to recognize because it allows you to simplify expressions that seem complex at first. In the original expression, once you identify
  • \((2x+1)^2\) as one square,
  • and \(y^2\) as another,
you can write it in the form \((2x+1)^2 - y^2\). This can then be factored using the difference of squares formula into \((2x+1-y)(2x+1+y)\). It's straightforward once you spot the structure, making it a reliable tool in algebraic simplification.
Expression Simplification
Expression simplification in algebra is about making expressions easier to understand and work with. After factoring an expression, it's ideal to check if it can be simplified further. Sometimes, expressions cannot be reduced any more, as is the case with the exercise
  • The expression \((2x+1-y)(2x+1+y)\)
was the result of successfully applying both perfect square trinomial and difference of squares techniques. Simplification often involves canceling terms, combining like terms, or finding common factors. However, not every expression can be distilled further. Learning when to stop is equally important, so always review if additional simplification is possible, but recognize when you鈥檝e achieved the simplest form.

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Most popular questions from this chapter

Factor. $$6 m^{2}+19 m+3$$

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