Problem 51 - A Graphical Approach to College Algebra

Question

Use the concepts of this section. Determine whether the description of the polynomial function \(P(x)\) with real coefficients is possible or not possible. (a) \(P(x)\) is of degree 3 and has zeros of \(1,2,\) and \(1+i\). (b) \(P(x)\) is of degree 4 and has four nonreal complex zeros. (c) \(P(x)\) is of degree 5 and \(-6\) is a zero of multiplicity 6. (d) \(P(x)\) has \(1+2 i\) as a zero of multiplicity 2.

Step-by-Step Solution

Answer

(a) Not possible, (b) Not possible, (c) Not possible, (d) Possible.

Step 1: Review the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that a polynomial of degree \(n\) has \(n\) roots, counting multiplicities, in the complex number system. If the coefficients are real, any complex zeros must occur in conjugate pairs.

Step 2: Analyze case (a)

A cubic polynomial of degree 3, such as \(P(x)\), can have 3 zeros. The zeros given are \(1, 2,\) and \(1+i\). Since the coefficients are real, the zero \(1+i\) implies that its conjugate \(1-i\) is also a zero. This makes a total of 4 distinct zeros (\(1, 2, 1+i, 1-i\)), which is not possible for a polynomial of degree 3. Thus, this case is not possible.

Step 3: Analyze case (b)

For a quartic polynomial of degree 4, like \(P(x)\), having four non-real complex zeros would require all zeros to occur in conjugate pairs because of the real coefficients. Since there are 4 non-real complex zeros and no zeros exist to balance as real roots or self-conjugate, it is impossible to have 4 non-real zeros. This case is not possible.

Step 4: Analyze case (c)

The polynomial \(P(x)\) is said to be of degree 5 and has \(-6\) as a zero with multiplicity 6. The multiplicity of a root is the number of times it appears as a factor in the polynomial. If \(-6\) has multiplicity 6, then the polynomial must be at least degree 6. Thus, it cannot be degree 5, making this scenario not possible.

Step 5: Analyze case (d)

A polynomial with real coefficients having a complex zero \(1+2i\) means the conjugate \(1-2i\) must also be a zero. If \(1+2i\) is a zero with multiplicity 2, this implies \(1+2i\) appears twice, and \(1-2i\) must also appear with multiplicity 2. Given this scenario, the polynomial must have at least degree 4 to accommodate these conditions. This is possible if the degree of the polynomial is 4 or higher; hence, it is possible.

Key Concepts

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is a key concept in understanding polynomial functions. It states that any non-zero polynomial equation of degree \(n\) will have exactly \(n\) roots in the complex number system. These roots include all complex numbers and take their multiplicities into account. This means if any root appears more than once, it is counted multiple times.
For example, consider a polynomial of degree 3. Based on the Fundamental Theorem of Algebra, it must have exactly 3 roots in total. These roots can be real numbers or complex numbers.
An important aspect to remember is that if a polynomial has real coefficients, any complex roots must appear as conjugate pairs. A complex conjugate of a root \(a + bi\) would be \(a - bi\). Therefore, having one complex root automatically ensures that its conjugate is also a root. This principle helps us ensure the context of polynomial roots fits appropriately within the structure set by real coefficients.

Complex Zeros and Conjugates

When dealing with polynomial functions with real coefficients, understanding the nature of complex zeros is crucial. A complex zero is typically of the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit.
Due to the nature of real coefficients, if a polynomial has a complex zero, its conjugate must also be a zero of the polynomial. So, if \(a + bi\) is a zero, then \(a - bi\) must be present as well. This is because the product of a complex number and its conjugate generates a real number, preserving the real nature of the polynomial's coefficients.

• If a polynomial is described as having complex zeros, always check to ensure that they occur in conjugate pairs.
• Conjugate pairs help to maintain the balanced nature of equations formed by polynomial roots, keeping the equation applicable with real coefficients.

Polynomial Degree and Zeros

The degree of a polynomial gives us vital information about the number of roots it should have. A polynomial of degree \(n\) can have up to \(n\) roots. Each root accounts for all multiplicities, whether they are real or complex.

• If a polynomial is claimed to have more zeros than its degree, there could be a misinterpretation or error.
• The multiplicity of a root also plays a crucial role. If a root has a multiplicity greater than one, it means the root appears more than once in factor form in the polynomial.
• Understanding the relation between the degree of a polynomial and the count and nature of its zeros guides us in solving and verifying polynomial equations.

For instance, if a polynomial is stated to have a degree 5, then having a root with multiplicity 6 isn't feasible, indicating an inconsistency. Always ensure that the sum of all multiplicities of zeros equals the polynomial's degree.