Problem 39 - Algebra and Trigonometry

Question

Find a polynomial with integer coefficients that satisfies the given conditions. \(P\) has degree 3 and zeros 2 and \(i\)

Step-by-Step Solution

Answer

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

Step 1: Determine Roots

Since the polynomial has degree 3 and the roots include 2 and \( i \), we need to find the third root. Complex roots appear as conjugate pairs in polynomials with real coefficients. Therefore, the third root must be \( -i \). The roots are 2, \( i \), and \( -i \).

Step 2: Write the Polynomial using Roots

The polynomial can be written as the product of factors corresponding to its roots. Since the roots are 2, \( i \), and \( -i \), the polynomial expression is \[ P(x) = (x - 2)(x - i)(x + i) \]

Step 3: Simplify using the Difference of Squares

Notice that \( (x - i)(x + i) \) is a difference of squares. Simplify this expression:\[ (x - i)(x + i) = x^2 - (i)^2 = x^2 + 1 \]Thus, the polynomial becomes:\[ P(x) = (x - 2)(x^2 + 1) \]

Step 4: Expand the Polynomial

Distribute \((x - 2)\) across \((x^2 + 1)\):\[ P(x) = x(x^2 + 1) - 2(x^2 + 1) \]The expansion is:\[ P(x) = x^3 + x - 2x^2 - 2 \] Combining like terms gives:\[ P(x) = x^3 - 2x^2 + x - 2 \]

Key Concepts

Integer Coefficients

Polynomials with integer coefficients are those where all the numbers in front of the variable terms are integers. This is crucial when constructing a polynomial from given roots, especially when one or more of the roots are complex numbers. For a polynomial to have integer coefficients, any non-real roots must come in conjugate pairs.

• This ensures that when multiplied, the imaginary parts cancel each other out.
• For example, if a polynomial has a root of \( i \), it must also have \( -i \) as a root to maintain real coefficients after expansion.
• This is because \[ (x - i)(x + i) = x^2 + 1, \] effectively removing any imaginary components and leaving integer coefficients behind.

Thus, when asked to find a polynomial with integer coefficients, ensuring complex conjugate pairs is a critical step.

Complex Conjugate Roots

Complex conjugate roots are key in polynomials with real and integer coefficients. They come in pairs of the form \( a + bi \) and \( a - bi \). This is because

• when multiplied, the imaginary numbers cancel each other out.
• This leads to terms that are entirely real.

For instance, in the problem we're examining, the presence of \( i \) automatically implies \( -i \) must also be a root.
This rule ensures that when expanded, the polynomial will have only real numbers (and potentially integer) coefficients. By employing this principle, instead of worrying about complex numbers, you merely expand the polynomial terms derived from these conjugate pairs.
In our solution, this is demonstrated by the term \[(x - i)(x + i) = x^2 + 1,\] showcasing that the multiplication cancels the complex parts.

Degree of Polynomial

The degree of a polynomial is a fundamental concept determining the polynomial's behavior and shape. It is the highest power of the variable in the polynomial expression.

• In our polynomial, the degree is 3, as indicated by the problem statement.
• The degree can also suggest the number of possible roots the polynomial could have.

For a degree 3 polynomial such as our example \[P(x) = x^3 - 2x^2 + x - 2,\] you typically expect three roots (real or complex). The degree provides a framework for constructing polynomials based on given roots because it informs us of how many factors need to be included in the polynomial equation.
Additionally, understanding the degree also gives insight into how the graph of the polynomial will behave, particularly the number of turning points and intercepts on an axis. This highlights the importance of confirming the degree when given partial root information to ensure all roots are accounted for and factored in accurately in the polynomial.