Express p(x)=x²-√29 as a product of linear factors.
Select one:
a. (x-2√29)(x-29)
b. (x-√29)(x+√29)
c. (x-29)(x+29)
d. (x-29)(x-29)
Answer (1)
Recognize the polynomial as a difference of squares.
Apply the difference of squares factorization: a 2 − b 2 = ( a − b ) ( a + b ) .
Factor x 2 − 29 as ( x − 4 29 ) ( x + 4 29 ) .
If the problem was x 2 − 29 , the factorization would be ( x − 29 ) ( x + 29 ) .
Explanation
Understanding the Problem We are given the polynomial p ( x ) = x 2 − 29 and asked to express it as a product of linear factors. This means we want to write it in the form ( x − a ) ( x − b ) for some constants a and b .
Recognizing Difference of Squares We can recognize that p ( x ) is in the form of a difference of squares, x 2 − ( 29 ) 2 . Recall that the difference of squares factorization is a 2 − b 2 = ( a − b ) ( a + b ) .
Applying Difference of Squares Factorization Applying the difference of squares factorization to p ( x ) , we have: x 2 − 29 = ( x − 4 29 ) ( x + 4 29 ) However, this is not one of the options. Let's re-examine the problem. We are given p ( x ) = x 2 − 29 . We want to find a and b such that x 2 − 29 = ( x − a ) ( x + a ) = x 2 − a 2 . Thus, we need a 2 = 29 , so a = 4 29 . The correct factorization should be ( x − 4 29 ) ( x + 4 29 ) .
Checking the Options However, looking at the options again, we see that option b is ( x − 29 ) ( x + 29 ) . If we expand this, we get x 2 − ( 29 ) 2 = x 2 − 29 , which is not the original polynomial. There seems to be a typo in the problem. The problem should be p ( x ) = x 2 − 29 . In this case, x 2 − 29 = ( x − 29 ) ( x + 29 ) .
Final Answer Assuming the problem was intended to be p ( x ) = x 2 − 29 , the correct factorization is ( x − 29 ) ( x + 29 ) .
Examples
The difference of squares factorization is a fundamental concept in algebra and is used in various applications. For example, it can be used to simplify algebraic expressions, solve equations, and analyze the behavior of functions. In physics, it can be used to analyze the motion of objects or the behavior of waves. For instance, consider the expression x 2 − 9 . We can factor this as ( x − 3 ) ( x + 3 ) . This factorization can help us find the roots of the equation x 2 − 9 = 0 , which are x = 3 and x = − 3 .
