Factor: $x^2+16$
A. $(x+4)(x-4)$
B. $(x-4)(x-4)$
C. $(x+4)(x+4)$
D. Cannot be factored
Answer (1)
The expression is x 2 + 16 .
Attempt to factor using real numbers: Not possible.
Attempt to factor using complex numbers: ( x + 4 i ) ( x − 4 i ) .
Since we are looking for factors with real numbers, the expression cannot be factored. The final answer is Cannot be factored .
Explanation
Understanding the Problem We are asked to factor the expression x 2 + 16 . This is a sum of squares, and we need to determine if it can be factored using real numbers or complex numbers.
Factoring with Real Numbers Let's first consider factoring with real numbers. We are looking for two numbers that multiply to 16 and add to 0 (since the coefficient of the x term is 0). However, there are no such real numbers. The factors of 16 are 1 and 16, 2 and 8, and 4 and 4. None of these pairs can add up to 0. Therefore, x 2 + 16 cannot be factored using real numbers.
Factoring with Complex Numbers Now, let's consider factoring with complex numbers. We can rewrite the expression as a difference of squares: x 2 − ( − 16 ) . Since − 1 = i 2 , we have x 2 − ( 16 i 2 ) = x 2 − ( 4 i ) 2 . Now we can use the difference of squares formula, a 2 − b 2 = ( a + b ) ( a − b ) , where a = x and b = 4 i . Thus, x 2 − ( 4 i ) 2 = ( x + 4 i ) ( x − 4 i ) .
Checking the Options However, the given options are: ( x + 4 ) ( x − 4 ) which equals x 2 − 16 ( x − 4 ) ( x − 4 ) which equals x 2 − 8 x + 16 ( x + 4 ) ( x + 4 ) which equals x 2 + 8 x + 16 Cannot be factored Since we are considering only real numbers, the expression x 2 + 16 cannot be factored using real numbers.
Final Answer Therefore, the correct answer is 'Cannot be factored'.
Examples
Factoring expressions like x 2 + 16 might seem abstract, but it's a fundamental concept in algebra with real-world applications. For example, in electrical engineering, impedance calculations often involve complex numbers and factoring similar expressions helps simplify circuit analysis. Also, in physics, when dealing with oscillatory motion or wave phenomena, factoring polynomials can aid in finding natural frequencies or modes of vibration. These applications highlight how mastering factoring extends beyond textbook problems, providing tools for solving practical engineering and physics challenges.
