Evaluate (x² + 19)/(2 - x) for x = 3i.

Verified step by step guidance

1. Substitute x = 3i into the given expression (x^2 + 19)/(2 - x). This means replacing every occurrence of x with 3i.
2. Simplify the numerator x^2 + 19. First, calculate (3i)^2, which involves squaring the imaginary number 3i. Recall that i^2 = -1, so (3i)^2 = 9i^2 = 9(-1) = -9. Add this result to 19 to simplify the numerator.
3. Simplify the denominator 2 - x. Replace x with 3i, so the denominator becomes 2 - 3i.
4. Combine the simplified numerator and denominator into the fraction. The expression now takes the form of a complex fraction with a real part and an imaginary part.
5. If necessary, rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator (2 + 3i). This step eliminates the imaginary part in the denominator, leaving a simplified complex number.

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i', which is defined as the square root of -1. In this question, '3i' is a purely imaginary number, which means its real part is zero.

Polynomial Evaluation

Polynomial evaluation involves substituting a specific value into a polynomial expression to compute its value. In this case, the expression (x^2 + 19) is a polynomial, and we need to evaluate it by substituting x with 3i, which requires calculating (3i)^2 and then adding 19.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In this question, the expression (x^2 + 19)/(2 - x) is a rational function, and evaluating it at x = 3i involves calculating both the numerator and the denominator separately before performing the division.