Evaluate x²+19 / 2−x for x = 3i.

Verified step by step guidance

1. Substitute \( x = 3i \) into the expression \( \frac{x^2 + 19}{2 - x} \).
2. Calculate \( x^2 \) where \( x = 3i \). Recall that \( i^2 = -1 \), so \( (3i)^2 = 9i^2 = 9(-1) = -9 \).
3. Substitute \( x^2 = -9 \) into the numerator: \( x^2 + 19 = -9 + 19 \).
4. Simplify the numerator: \( -9 + 19 = 10 \).
5. Substitute \( x = 3i \) into the denominator: \( 2 - x = 2 - 3i \).

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 imaginary part multiplied by the imaginary unit 'i', which is defined as the square root of -1. Understanding complex numbers is essential for evaluating expressions that involve imaginary units, such as substituting x = 3i in the given expression.

Polynomial Evaluation

Polynomial evaluation involves substituting a specific value into a polynomial expression to compute its value. In this case, the expression x² + 19 / (2 - x) is a polynomial that needs to be evaluated at x = 3i. This process requires careful handling of both the real and imaginary components of the expression.

Division of Complex Numbers

When dividing complex numbers, it is important to simplify the expression by multiplying the numerator and denominator by the conjugate of the denominator. This helps eliminate the imaginary unit from the denominator, making the expression easier to interpret. In the context of the given question, understanding how to handle division involving complex numbers is crucial for arriving at the correct result.