Question

Find a polynomial with real coefficients that has the given zeros. (There are many correct answers.) $$2,2,2,4 i,-4 i$$

Short answer

The polynomial with real coefficients that has roots of 2 (of multiplicity 3), 4i and -4i is \((x-2)^3(x^2 + 16)\).

Step-by-step solution

Treat the Real Roots

Firstly, treat the real roots. The polynomial (x - 2) accommodates the real root 2. Since 2 is a root of multiplicity 3, this feature would be represented as \((x - 2)^3\). So, the polynomial accommodating real roots is \((x-2)^3\).

Treat the Imaginary Roots

Next, deal with the complex roots. The roots here are 4i and -4i. It's important to remember that in a polynomial with real coefficients, non-real roots always come in conjugate pairs. In this case, the pair would give \((x-4i)(x+4i)\), and multiplying these would lead to the quadratic \(x^2 + 16\). Therefore, the polynomial accommodating the complex roots is \(x^2 + 16\).

Combine the polynomials

Finally, assemble the polynomial that accommodates the real roots and the polynomial that accommodates the non-real roots together to form the final polynomial. This means multiplying \((x-2)^3\) and \((x^2 + 16)\). The final polynomial is \((x-2)^3(x^2 + 16)\).

Key concepts

Multiplicity of Roots

In polynomials, roots can sometimes appear more than once. This repetition is referred to as the "multiplicity" of a root. Simply put, if a root appears multiple times, it affects the structure of the polynomial.
For example, when we have the root 2 with a multiplicity of 3, it means the factor ing the root appears thrice as \((x - 2)^3\).

• If a root has a multiplicity of 1, it crosses the x-axis.
• If the multiplicity is even, the graph only touches the x-axis but doesn't cross it.
• If the multiplicity is odd, it will cross the axis.

This affects the shape and behavior of the polynomial's graph. In our exercise, the root 2 appears three times, meaning it has a multiplicity of 3 and is represented as \((x-2)^3\) in the polynomial.

Complex Conjugate Roots

When polynomials have real coefficients and non-real roots, these non-real roots must come in conjugate pairs. This is due to the complex conjugate root theorem which states if \(a+bi\) is a root, then \(a-bi\) must also be a root.
In the given exercise, the non-real roots are \(4i\) and \(-4i\). These form a pair of complex conjugates. When you multiply these conjugate pairs, you get a polynomial expression with real coefficients. The conjugate multiplication results in \((x - 4i)(x + 4i) = x^2 + 16\).

• This quadratic expression has no real part, only an imaginary component squared, eliminating the imaginary component altogether.
• The resulting polynomial still fits within the real coefficient requirement.

Thus, complex conjugate roots allow the formation of real-coefficient polynomials even if they initially contain imaginary numbers.

Real and Imaginary Roots

In this exercise, the distinction between real and non-real (imaginary) roots is crucial. Real roots are straightforward; they are simply numbers without imaginary components. Non-real roots involve the imaginary unit \(i\), where \(i^2 = -1\).
Real roots in our exercise include the number 2, repeated thrice, while the imaginary roots are \(4i\) and \(-4i\).

• Real roots often result in linear factors of the polynomial.
• Imaginary roots, when in pairs, result in quadratic factors.

When finding polynomials with these roots, it’s necessary to treat them differently: real roots as repeated factors and imaginary roots as conjugate pairs. The combination process eventually results in a complete polynomial form that encompasses both root types.

Factoring Polynomials

Factoring polynomials involve expressing them as a product of simpler polynomials, often linear or quadratic. Identifying and using roots helps in this factoring process.
In our problem, we have real and imaginary roots to consider:

• Real roots like 2 with multiplicity are written as \((x-2)^3\). This expression captures the repetition of the root.
• Imaginary roots, such as \(4i\) and \(-4i\), are written as conjugate pairs, resulting in \((x^2 + 16)\).

By factoring polynomials, you rewrite them in ways that reveal possible graph shapes or simplify calculation. Eventually, this gives us a full polynomial: \((x-2)^3 (x^2 + 16)\). This approach not only includes all roots correctly but also ensures the polynomial maintains real number coefficients.