Difference of Squares gives which complex factors for the expression [tex]x^2+3[/tex]?
A. [tex](x+3 i)(x-3 i)[/tex]
B. [tex](x-i \sqrt{3})(x-i \sqrt{3})[/tex]
C. [tex](x+i \sqrt{3})(x-i \sqrt{3})[/tex]
D. [tex](x+3 i)^2(x-3 i)^2[/tex]
Answer (2)
The complex factors of the expression x 2 + 3 can be found using the difference of squares method. The correct factorization is ( x + i 3 ) ( x − i 3 ) , which corresponds to option C. Thus, the answer is C.
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Rewrite the expression x 2 + 3 as x 2 − ( − 3 ) .
Express − 3 as ( i 3 ) 2 .
Apply the difference of squares factorization: x 2 − ( i 3 ) 2 = ( x + i 3 ) ( x − i 3 ) .
The complex factors are ( x + i 3 ) ( x − i 3 ) .
Explanation
Understanding the Problem We are given the expression x 2 + 3 and asked to find its complex factors using the difference of squares. The difference of squares factorization is a 2 − b 2 = ( a + b ) ( a − b ) . We need to rewrite x 2 + 3 in the form a 2 − b 2 using complex numbers.
Rewriting the Expression To use the difference of squares, we need to rewrite the expression x 2 + 3 as a difference. We can rewrite it as x 2 − ( − 3 ) .
Expressing -3 as a Square Now we need to express − 3 as a square of a complex number. We know that i 2 = − 1 , so we can write − 3 = 3 i 2 = ( i 3 ) 2 .
Applying Difference of Squares So, we can rewrite the expression as x 2 − ( i 3 ) 2 . Now we can apply the difference of squares factorization: x 2 − ( i 3 ) 2 = ( x + i 3 ) ( x − i 3 ) .
Final Answer Comparing our result with the given options, we see that the correct factorization is ( x + i 3 ) ( x − i 3 ) . Therefore, the answer is C.
Examples
The difference of squares factorization is a powerful tool not only in algebra but also in various fields like physics and engineering. For instance, in electrical engineering, when analyzing alternating current (AC) circuits, complex numbers are used to represent impedances. Factoring expressions involving impedances using the difference of squares can simplify circuit analysis and design, helping engineers to understand the behavior of AC circuits more effectively. This technique allows for easier calculations of current, voltage, and power in complex circuits, ensuring efficient and reliable performance.
