Problem 36 - Algebra and Trigonometry

Question

Find a polynomial with integer coefficients that satisfies the given conditions. \(P\) has degree 2 and zeros \(1+i \sqrt{2}\) and \(1-i \sqrt{2}\)

Step-by-Step Solution

Answer

The polynomial is \( P(x) = x^2 - 2x + 3 \).

Step 1: Identify the Zeros

Since the polynomial has zeros at \( 1+i\sqrt{2} \) and \( 1-i\sqrt{2} \), we recognize that \(P(x)\) can be constructed as \( (x - (1+i\sqrt{2}))(x - (1-i\sqrt{2})).\) These zeros are complex conjugates, which ensures they satisfy the condition for having real coefficients.

Step 2: Expand the Expression

To find \( P(x) \), multiply out the expression: \( (x - (1+i\sqrt{2}))(x - (1-i\sqrt{2})) \). Using the difference of squares formula, this becomes: \[(x - 1 - i\sqrt{2})(x - 1 + i\sqrt{2}) = ((x-1) - i\sqrt{2})((x-1) + i\sqrt{2})\]\[= (x-1)^2 - (i\sqrt{2})^2\]\[\]Since \((i\sqrt{2})^2 = -2\), the expression simplifies to: \[(x-1)^2 + 2\].

Step 3: Simplify the Polynomial

Expand \((x-1)^2\):\[(x-1)^2 = x^2 - 2x + 1\] Now substitute this back into the polynomial expression:\[P(x) = (x^2 - 2x + 1) + 2\]Thus, simplifying gives:\[P(x) = x^2 - 2x + 3\].

Step 4: Verify Integer Coefficients

Check that all coefficients of the polynomial \( P(x) = x^2 - 2x + 3 \) are integers. The coefficients are 1, -2, and 3, all of which are integers, satisfying the problem's requirement.

Key Concepts

Complex Conjugates

When you have complex numbers, like the zeros given in this exercise, complex conjugates come in handy. A complex conjugate is just a pair of complex numbers that have the same real part but opposite imaginary parts. For example, the complex conjugate of \( 1 + i\sqrt{2} \) is \( 1 - i\sqrt{2} \).
This means that when you know one zero of a polynomial, its complex conjugate is often the other zero, as seen in this exercise.

• A polynomial with real coefficients will always have its non-real zeros come in complex conjugate pairs.
• This ensures that when factored and simplified, the polynomial remains with real and possibly integer coefficients.

Thus, complex conjugates help maintain the polynomial's integrity when it needs to have real coefficients. Multiplying conjugates results in eliminating the imaginary parts, contributing towards a polynomial with real numbers.

Degree of Polynomial

Understanding the degree of a polynomial is crucial when solving such exercises. The degree of a polynomial is the highest power of the variable in the polynomial expression.
In the problem above, we identified it as degree 2.

• For this polynomial, \( P(x) = x^2 - 2x + 3 \), the degree is 2, which is the highest power of \( x \) (i.e., \( x^2 \)).
• The degree of a polynomial influences both its shape on a graph and the number of solutions or zeros it has.

Having a degree of 2 tells us that there will be exactly two zeros provided the polynomial is fully factored and real, complex numbers might be involved. This value sets a balance in the polynomial, guiding how we solve or analyze it.

Integer Coefficients

Polynomials with integer coefficients are a common requirement in many exercises, including the original problem in this exercise. Integer coefficients are simply the numbers that multiply the terms of the polynomial, and they are whole numbers.

• In our solution, \( P(x) = x^2 - 2x + 3 \), the coefficients are 1, -2, and 3.
• Ensuring integer coefficients means the solutions are neat, avoiding fractions or decimals.

This property provides a polynomial that is easier to manage and verify, especially in theoretical and practical applications in mathematics. It's a straightforward way of making the polynomial more robust and suitable for various problems.