Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a polynomial function that has the given zeros. (There are many correct answers.) \(1+\sqrt{2}, 1-\sqrt{2}\)

Step-by-Step Solution

Answer

The polynomial function that has the given zeros (\(1+\sqrt{2}\), \(1-\sqrt{2}\)) is \(x^2 - 2x - 1\).

Step 1: Write the factor of the polynomial

Step 1 includes identifying the factors of the polynomial. The given zeros are \(a + bi\) and \(a - bi\). The factors associated with these zeros would be \((x - (a + bi))\) and \((x - (a - bi))\). Substituting the given zeros we get \((x - (1+\sqrt{2}))\) and \((x - (1-\sqrt{2}))\)

Step 2: Multiply the factors

The polynomial can be attained by multiplying the factors together. Multiply \((x - (1+\sqrt{2})) (x - (1-\sqrt{2}))\). The multiplication results in \(x^2 - 2x + 1 - 2\)

Step 3: Simplify the polynomial

Simplify the polynomial from step 2. The final polynomial is \(x^2 - 2x - 1\)

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra that involves breaking down a polynomial into simpler 'factor' polynomials whose product is the original polynomial. These factors are generally easier to work with and can help simplify complex algebraic operations.

When given a set of zeros, such as \(1+\sqrt{2}\) and \(1-\sqrt{2}\), we can find the corresponding factors of the polynomial. For each zero \(a\), there's a corresponding factor of \(x - a\). In this case, we have two factors: \(x - (1+\sqrt{2})\) and \(x - (1-\sqrt{2})\).

To find the polynomial, you multiply these factors together. This process turns the problem of finding a polynomial with given zeros into a multiplication exercise, which is generally a more straightforward operation.

The beauty of factoring lies in its ability to make complex problems more manageable; and in the context of zeros, it aids in constructing the original polynomial with ease.

Complex Conjugate Zeros

When dealing with polynomial functions, there's an interesting occurrence known as complex conjugate zeros. They come in pairs, like the duo \(1+\sqrt{2}\) and \(1-\sqrt{2}\), which are not complex numbers themselves but illustrate the concept well since they involve a radical expression similar to how complex numbers involve an imaginary unit.

If a polynomial has real coefficients, any complex zero such as \(a + bi\) will always have a conjugate pair \(a - bi\), ensuring the product of the factors will result in a polynomial with real coefficients.

In the context of our exercise, we don't have an imaginary part, but the principle of conjugate pairs still provides an important foundation. When you find zeros that are related in this way, you can always multiply the factors to get a polynomial with real coefficients, which in our case, results in simple conjugates without the imaginary part. Recognizing complex conjugates or their simpler counterparts is vital in simplifying the process of creating polynomials.

Simplifying Polynomials

Once the factors of the polynomial are determined, we need to multiply them to find the polynomial in its expanded form. However, the product of these factors often results in a longer expression that can be simplified.

Simplifying polynomials involves combining like terms and reducing the expression to its most compact form. In our solution, after multiplying \(x - (1+\sqrt{2})) (x - (1-\sqrt{2}))\), we get \(x^2 - 2x + 1 - 2\).

The next step is to simplify by combining similar terms. For the expression \(x^2 - 2x + 1 - 2\), we combine the constant terms to get \(x^2 - 2x - 1\). The simplification gives us a clearer and more concise polynomial function, which is easier to analyze and work with in algebraic problems.

Understanding how to simplify polynomials correctly is essential for algebra students as it makes interpreting and working with polynomial functions much more intuitive.