Problem 27 - Algebra for College Students

Question

Find a polynomial equation with real coefficients that has the given roots. $$0, i \sqrt{2}$$

Step-by-Step Solution

Answer

The polynomial equation is \( x^3 + 2x \)

Step 1 - Identify the given roots

The roots given are 0 and \( i \sqrt{2} \)

Step 2 - Include the conjugate

Since the polynomial has real coefficients, the complex root must have its conjugate as well. The conjugate of \( i \sqrt{2} \) is \( -i \sqrt{2} \)

Step 3 - Form the polynomial factors

Each root gives a factor of the polynomial. The roots 0, \( i \sqrt{2} \), and \( -i \sqrt{2} \) give the factors \( x \), \( x - i \sqrt{2} \), and \( x + i \sqrt{2} \)

Step 4 - Combine and multiply

Combine and multiply the factors to form the polynomial. First, combine \( (x - i \sqrt{2}) \) and \( (x + i \sqrt{2}) \): Using the difference of squares formula: \( (x - i \sqrt{2})(x + i \sqrt{2}) = x^2 - (i \sqrt{2})^2 = x^2 + 2 \) Then multiply with the remaining factor, \( x \): \( x(x^2 + 2) = x^3 + 2x \)

Key Concepts

real coefficients

A polynomial equation with real coefficients means that all the numbers in the polynomial (the coefficients) are real numbers, not imaginary or complex ones. For example, in the polynomial equation \(x^3 + 2x\), the numbers 1 and 2 are all real. For a polynomial with real coefficients, if it has any complex roots, those roots must occur in conjugate pairs. That means if \(a + bi\) is a root, then \(a - bi\) must also be a root. Having real coefficients ensures the polynomial's graph intersects the real axis in a ‘balanced’ way. This is why knowing about real coefficients is important for understanding how the polynomial behaves.

complex conjugate roots

Complex conjugate roots are pairs of complex numbers that are mirror images with respect to the real axis. If we have a complex root such as \( i \sqrt{2} \), its conjugate is \( -i \sqrt{2} \). The conjugate of a complex number is formed by changing the sign of the imaginary part. If a polynomial with real coefficients has a complex root, the conjugate root must also be present to maintain real coefficients. This concept is essential because it helps in forming polynomial equations correctly. For instance, if a polynomial equation has the roots 0, \( i \sqrt{2} \), then \( -i \sqrt{2} \) must also be a root.

difference of squares

The difference of squares formula is used to simplify expressions that look like \((a - b)(a + b)\). The formula is \( a^2 - b^2 \). This is crucial in polynomial multiplication and factoring. In our example, we have \((x - i \sqrt{2})(x + i \sqrt{2})\). Applying the difference of squares formula, we get \((x)^2 - (i \sqrt{2})^2\). Simplifying further, \((i \sqrt{2})^2 = -2\), so \(x^2 - (-2)\), which equals \(x^2 + 2\). This process helps in reducing complex expressions to simpler, usable polynomials.

polynomial multiplication

Polynomial multiplication is a process of multiplying two or more polynomials to get a single polynomial. Each term in the first polynomial is multiplied by each term in the second polynomial. After that, the products are added together. In the step by step solution, we first find the product of the two complex conjugate factors: \((x - i \sqrt{2})(x + i \sqrt{2}) = x^2 + 2\). Then, we multiply this result by the remaining factor \(x\). So, \(x(x^2 + 2)\) results in \(x^3 + 2x\). This multiplication process combines all the factors of the polynomial to give us the final polynomial equation which encompasses all the given roots.