Evaluate x² − 2x + 2 for x = 1 + i.

Verified step by step guidance

1. Substitute $x=1+i$ into the expression $x^2-2x+2$.
2. Calculate $(1+i{)}^2$ using the formula $(a+b{)}^2=a^2+2ab+b^2$.
3. Simplify $(1+i{)}^2$ to get $1+2i+i^2$.
4. Remember that $i^2=-1$, so replace $i^2$ with $-1$ in the expression.
5. Substitute the simplified $(1+i{)}^2$ and $-2(1+i)$ into the original expression and simplify.

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. Understanding complex numbers is essential for evaluating expressions that include imaginary components, such as the one in this question.

Polynomial Evaluation

Polynomial evaluation involves substituting a specific value into a polynomial expression to compute its value. In this case, we need to substitute x = 1 + i into the polynomial x² − 2x + 2, which requires performing operations like addition, multiplication, and squaring complex numbers.

Algebraic Operations with Complex Numbers

Performing algebraic operations with complex numbers requires following the same rules as with real numbers, while also applying the property that i² = -1. This includes addition, subtraction, multiplication, and division, which are crucial for simplifying expressions and obtaining the final result when evaluating polynomials with complex inputs.