Factor this polynomial completely.
$x^2-49$
A. $(x-7)(x-7)$
B. $(x+49)(x-49)$
C. $(x-49)(x-49)$
D. $(x+7)(x-7)$
Answer (1)
The polynomial x 2 − 49 is a difference of squares. We use the formula a 2 − b 2 = ( a + b ) ( a − b ) to factor it. In this case, x 2 − 49 = ( x + 7 ) ( x − 7 ) . The final answer is ( x + 7 ) ( x − 7 ) .
Explanation
Recognizing the Pattern We are asked to factor the polynomial x 2 − 49 completely. This looks like a difference of squares, which has a specific factorization pattern.
Applying the Formula The difference of squares factorization formula is a 2 − b 2 = ( a + b ) ( a − b ) . In our case, we have x 2 − 49 , which can be written as x 2 − 7 2 . So, a = x and b = 7 .
Factoring the Polynomial Applying the formula, we get x 2 − 49 = ( x + 7 ) ( x − 7 ) .
Selecting the Correct Option Comparing our result with the given options, we see that option D, ( x + 7 ) ( x − 7 ) , matches our factorization.
Examples
Factoring polynomials, like x 2 − 49 , is useful in many areas. For example, if you're designing a rectangular garden and you know the area can be represented by x 2 − 49 , factoring it into ( x + 7 ) ( x − 7 ) tells you the possible dimensions of the garden. This helps in planning the layout and optimizing the space.
