Problem 42 - Algebra and Trigonometry

Question

Find a polynomial with integer coefficients that satisfies the given conditions. S has degree 4 and zeros 2\(i\) and 3\(i\).

Step-by-Step Solution

Answer

The polynomial is \(x^4 + 13x^2 + 36\).

Step 1: Identify the Roots

Given that the polynomial has zeros at 2i and 3i, we know that complex roots occur in conjugate pairs when coefficients are real. Therefore, the roots are 2i, -2i, 3i, and -3i.

Step 2: Create Linear Factors

For each root, there is a corresponding factor of the form (x - root). Thus, the factors are (x - 2i), (x + 2i), (x - 3i), and (x + 3i).

Step 3: Multiply Conjugate Pairs

Multiply the conjugate pairs: (x - 2i)(x + 2i) = x^2 + 4 (x - 3i)(x + 3i) = x^2 + 9

Step 4: Combine Factors

Combine the results from Step 3: (x^2 + 4)(x^2 + 9)

Step 5: Expand the Polynomial

Expand the expression from Step 4:\[(x^2 + 4)(x^2 + 9) = x^4 + 9x^2 + 4x^2 + 36 = x^4 + 13x^2 + 36\]

Step 6: Verify Degree and Coefficients

The polynomial, \(x^4 + 13x^2 + 36\), has a degree of 4 and integer coefficients, meeting the conditions of the exercise.

Key Concepts

Complex Roots

When dealing with polynomials, understanding complex roots is essential. Complex roots often appear in pairs known as conjugates. Conjugates are numbers like \(2i\) and \(-2i\), which share a real component but differ in the sign of their imaginary part. This pairing happens because polynomials with real coefficients must have real numbers as endpoints. Thus, every time you spot a complex root in a polynomial with real coefficients, its conjugate is also a root. This symmetry in pairing maintains a tidy balance of realness in the polynomial, allowing us to ensure that every complex quirk has a matching counterpart.

Degree of Polynomial

The degree of a polynomial is the highest power of its variable. It’s a straightforward but crucial identifier of a polynomial's complexity and the number of zeros it can have. In our case, the degree is 4, indicating that the polynomial will consist of terms going up to \(x^4\). The degree also tells us there will be 4 roots or zeros for this particular polynomial, taking into account both real and complex roots. So, when constructing a polynomial, knowing the degree helps you anticipate the structure and number of factors you will encounter when breaking down or building up the polynomial.

Linear Factors

Linear factors are the building blocks of a polynomial when represented in factored form. They appear as expressions like \((x - a)\), where \(a\) is a root of the polynomial. In the exercise, with roots at \(2i, -2i, 3i,\) and \(-3i\), we construct linear factors as \((x - 2i)\), \((x + 2i)\), \((x - 3i)\), and \((x + 3i)\). Each linear factor corresponds to a root and reflects how the polynomial relates to the x-axis graphically. By mapping each complex root and its conjugate pair into these factors, we create a framework that the polynomial can be built from.

Expanding Polynomials

Expanding a polynomial means transitioning from a factored form to a standard form, expressed as a sum of terms. This process involves multiplying the linear factors together. To illustrate, consider our polynomial's factors: \((x^2 + 4)\) and \((x^2 + 9)\). When these are expanded, you perform multiplication across all terms:

• Multiply each term in the first binomial by each term in the second.
• Combine like terms to simplify the expression.

For our exercise, this expansion led to \(x^4 + 13x^2 + 36\), transforming the polynomial into an easily readable form, while preserving its roots and satisfying the initial conditions of integer coefficients.