Which of the following represents the factorization of the binomial below?
[tex]x^2-81[/tex]
A. [tex](x+9)(x+9)[/tex]
B. [tex](x)(x-81)[/tex]
C. [tex](x+9)(x-9)[/tex]
D. [tex](x-9)(x-9)[/tex]
Answer (2)
The problem requires factoring the binomial x 2 − 81 .
Recognize x 2 − 81 as a difference of squares.
Apply the formula a 2 − b 2 = ( a + b ) ( a − b ) with a = x and b = 9 .
Factor x 2 − 81 into ( x + 9 ) ( x − 9 ) .
The correct factorization is ( x + 9 ) ( x − 9 ) .
Explanation
Recognizing the Difference of Squares We are asked to factor the binomial x 2 − 81 . This is a difference of squares, which can be factored using the formula a 2 − b 2 = ( a + b ) ( a − b ) .
Applying the Formula In our case, a = x and b = 9 , since x 2 − 81 = x 2 − 9 2 . Applying the difference of squares formula, we get x 2 − 81 = ( x + 9 ) ( x − 9 ) .
Selecting the Correct Option Comparing our result with the given options, we see that option C, ( x + 9 ) ( x − 9 ) , matches our factorization. Therefore, the correct answer is C.
Examples
The difference of squares factorization is useful in many areas, such as simplifying algebraic expressions and solving equations. For example, if you are designing a rectangular garden and want to know the dimensions that would give you a specific area, and the area can be expressed as a difference of squares, you can use this factorization to find the possible dimensions. Suppose the area of the garden is represented by x 2 − 25 , where x is related to the dimensions. Factoring this gives ( x + 5 ) ( x − 5 ) , which helps in determining the possible values for the dimensions of the garden.
The expression x 2 − 81 can be factored as ( x + 9 ) ( x − 9 ) , which is a result of recognizing it as a difference of squares. The correct option from the choices given is C: ( x + 9 ) ( x − 9 ) .
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