Problem 51 - College Algebra

Question

Evaluate \(x^{2}-2 x+2\) for \(x-1+i\)

Step-by-Step Solution

Answer

The final simplified expression after substituting the given complex value in the given expression is \(x^2 -2ix +3 - 2x - 2i + 3\)

Step 1: Substitution

Substitute \(x\) in the polynomial with \(x-1+i\) for \(x^{2}-2 x+2\). We will have: \((x-1+i)^{2}-2(x-1+i)+2\)

Step 2: Expand the Square and Distribute

Expand the square and distribute the real and imaginary parts. We will have: \((x-1)^{2} + 2(x-1)i + i^{2} + 2 - 2(x-1+i)\)

Step 3: Simplify

Simplify the expression, considering that \(i^{2} = -1\): \(x^2 -2x +1 +2xi -i^2 +2 -2x +2 -2i = x^2 -2ix +3\)

Step 4: Final Simplification

This expression can be further simplified to give the final answer. When we simplify the expression, we end up with: \(x^2 -2ix +3 -2x -2i +3\)

Key Concepts

Polynomial Substitution

Polynomial substitution is a fundamental technique in algebra that involves replacing a variable within a polynomial with another expression. This process is essential when evaluating polynomials at specific points, especially when those points involve complex numbers, as it was demonstrated in the given exercise.

Consider the polynomial expression \(x^2 - 2x + 2\). Substituting \(x\) with \(x-1+i\) requires us to plug the complex expression into every occurrence of \(x\) in the polynomial. Therefore, the modified polynomial becomes \( (x-1+i)^2 - 2(x-1+i) + 2 \). Substitution is the first step towards simplifying or evaluating a polynomial at a specific value or expression. The focus should be on careful replacement to avoid errors that may complicate later steps.

Complex Number Simplification

Complex number simplification involves reducing expressions involving complex numbers to their simplest form. This includes combining like terms, utilizing properties of the imaginary unit \(i\), and performing arithmetic operations such as addition, subtraction, and multiplication.

When simplifying \( (x-1+i)^2 - 2(x-1+i) + 2 \), the expression will expand and will include terms with real coefficients, imaginary coefficients, and possibly powers of \(i\). Since \(i^2 = -1\), any occurrence of \(i^2\) can be replaced with \( -1 \) to further simplify the expression. In the given solution, after expanding and distributing, the occurrence of \( i^2 \) was replaced and the expression was simplified by combining like terms, ultimately yielding the simpler expression \( x^2 -2ix +3 \).

Imaginary Unit Arithmetic

Imaginary unit arithmetic pertains to calculations involving the imaginary unit \(i\), defined by the property that \(i^2 = -1\). Understanding this property is crucial when working with complex numbers, as it underlies the simplification process of complex expressions.

In cases such as the given exercise, when expanding \( (x-1+i)^2 \) for example, one must apply the distributive property and then simplify the terms involving \(i\). The occurrence of \(i^2\) in the expanded form must be recognized as \( -1 \) to perform the correct simplification, converting what might initially appear to be an intimidating expression into one that is easier to manage. Correctly handling powers of \(i\), such as \(i^2\), \(i^3\), and beyond, is integral to working effectively with complex numbers in algebraic expressions.