Question 21 (5 points)
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Identify the factorization of $36+25 x^2$.
A. $(5+9 x i)(5-9 x i)$
B. $(5 x+6 i)(5 x-6 i)$
C. $(9 i-5 x)(9+5 x i)$
D. $(6+5 x)(6-5 x)$
Answer (1)
Rewrite the expression 36 + 25 x 2 as 6 2 − ( 5 x i ) 2 .
Apply the difference of squares factorization: ( 6 + 5 x i ) ( 6 − 5 x i ) .
Recognize that ( 5 x + 6 i ) ( 5 x − 6 i ) = 25 x 2 + 36 .
The correct factorization is ( 5 x + 6 i ) ( 5 x − 6 i ) .
Explanation
Understanding the Problem We are asked to identify the correct factorization of the expression 36 + 25 x 2 from the given options.
Rewriting the Expression We can rewrite the expression 36 + 25 x 2 as 6 2 − ( 5 x i ) 2 because i 2 = − 1 . This allows us to use the difference of squares factorization.
Applying Difference of Squares Using the difference of squares factorization, a 2 − b 2 = ( a + b ) ( a − b ) , where a = 6 and b = 5 x i , we have: 36 + 25 x 2 = 6 2 − ( 5 x i ) 2 = ( 6 + 5 x i ) ( 6 − 5 x i ) So the factorization is ( 6 + 5 x i ) ( 6 − 5 x i ) .
Identifying the Correct Option Comparing our result with the given options, we see that the correct factorization is ( 6 + 5 x i ) ( 6 − 5 x i ) , which can also be written as ( 6 + 5 x i ) ( 6 − 5 x i ) . However, we need to check if any of the given options are equivalent. Notice that ( 5 x + 6 i ) ( 5 x − 6 i ) = ( 5 x ) 2 − ( 6 i ) 2 = 25 x 2 − 36 i 2 = 25 x 2 + 36 . Therefore, the correct factorization is ( 5 x + 6 i ) ( 5 x − 6 i ) .
Final Answer Thus, the correct factorization of 36 + 25 x 2 is ( 5 x + 6 i ) ( 5 x − 6 i ) .
Examples
Complex numbers and their factorization are crucial in electrical engineering for analyzing AC circuits. For instance, when dealing with impedance, which includes resistance and reactance, complex numbers help simplify calculations. Factoring expressions involving complex numbers allows engineers to understand the behavior of circuits and design them effectively. This ensures that devices like smartphones and computers function as intended, relying on the precise manipulation of electrical signals using complex number principles.
