Problem 44 - Algebra and Trigonometry

Question

Find a polynomial with integer coefficients that satisfies the given conditions. \(U\) has degree \(5,\) zeros \(\frac{1}{2},-1,\) and \(-i,\) and leading coefficient \(4 ;\) the zero \(-1\) has multiplicity \(2 .\)

Step-by-Step Solution

Answer

The polynomial is \(8x^5 + 12x^4 + 4x^3 - 4x^2 - 4x - 4\).

Step 1: Identify the Known Information

The polynomial \( U(x) \) has degree 5, zeros \( \frac{1}{2}, -1 \) with multiplicity 2, and \(-i\). It also has a leading coefficient of 4. The complex zero \( -i \) implies \( i \) is also a zero due to complex conjugate pairs having real coefficients.

Step 2: Express Zeros as Factors

Each zero can be expressed as a factor of the polynomial. The zero \( \frac{1}{2} \) corresponds to the factor \((x - \frac{1}{2})\). The zero \(-1\) with multiplicity 2 corresponds to the factor \((x + 1)^2\). The zeros \( -i \) and \( i \) correspond to the factors \((x + i)\) and \((x - i)\).

Step 3: Construct the Polynomial Equation

The polynomial can be expressed using these factors: \(U(x) = a(x - \frac{1}{2})(x + 1)^2(x + i)(x - i)\), where \(a\) is the leading coefficient constant, which is 4.

Step 4: Combine Complex Conjugate Pairs

Combine the complex conjugate factors: \((x + i)(x - i) = x^2 + 1\). Now the polynomial is \(U(x) = 4(x - \frac{1}{2})(x + 1)^2(x^2 + 1)\).

Step 5: Clear the Fraction in the Factor

To eliminate the fraction, multiply \(x - \frac{1}{2}\) by 2. The factor becomes \((2x - 1)\). The polynomial is now \(U(x) = 4(2x - 1)(x + 1)^2(x^2 + 1)\).

Step 6: Expand the Polynomial

Begin expanding the polynomial by multiplying adjacent factors:1. Expand \((x + 1)^2 = x^2 + 2x + 1\).2. Mate \((2x - 1)\) with the expanded \((x + 1)^2\). Next, incorporate \((x^2 + 1)\) to get the polynomial form.

Step 7: Final Expansion and Expression

Expand the expression fully:\((2x - 1)(x^2 + 2x + 1) = 2x^3 + 4x^2 + 2x - x^2 - 2x - 1 = 2x^3 + 3x^2 - 1\).Then multiply \((2x^3 + 3x^2 - 1)\) by \(x^2 + 1\) to get the complete expression:\(2x^5 + 3x^4 - x^2 + 2x^3 + 3x^2 - x - 1\).Multiply by 4: \(8x^5 + 12x^4 + 4x^3 - 4x^2 - 4x - 4\).

Step 8: Verify the Polynomial

Check that the polynomial has a degree of 5 and that all conditions are met, including integer coefficients and its leading term being \(8x^5\).

Key Concepts

Integer Coefficients

When dealing with polynomials, it's often necessary for all the coefficients in the expression to be integers, especially in educational and certain applied contexts. Let's break down what this means and how to achieve it.
Integer coefficients mean that each term's coefficient in the polynomial is a whole number. This ensures easier computation and clear interpretation. In the original exercise, the polynomial is supposed to have integer coefficients. Start by acknowledging that while the given zero \(\frac{1}{2}\) manifests as a fraction, the factor \((x - \frac{1}{2})\) can be modified.

• To clear the fraction, we need to multiply the entire polynomial by the denominator of the fractional zero, which is \(2\).
• This transforms \((x - \frac{1}{2})\) into \((2x - 1)\), providing an integer coefficient of \(1\) for \(x\).

Adhering to the condition of integer coefficients aligns with practical usage and reflects the exercise's requirements.

Polynomial Degree

Understanding the degree of a polynomial is crucial, as it gives us insight into the shape and behavior of the curve. The degree of a polynomial is the highest power exponent of the variable \(x\). In the given exercise, the specified polynomial must have a degree of 5.

• The degree determines the number of zeros a polynomial can have, counting each zero by its multiplicity.
• In this task, we see a combination of zeros: \(\frac{1}{2}\) (once), \(-1\) (twice, due to multiplicity), and \(-i, i\) (forming a conjugate pair).

Consequently, these zeros imply the degree, as they satisfy the provided conditions.
When aiming for a polynomial of degree 5, as soon as all zeros are accounted for, including the multiplicities, the structure reflects the degree correctly. This ensures alignment with the polynomial’s behavior expected by its description.

Complex Zeros

Complex zeros are an advanced concept, but understanding them is essential for fully grasping polynomial functions. Complex zeros occur in conjugate pairs when polynomials have real coefficients.
In this exercise, one of the zeros is \(-i\), which inherently implies its conjugate \(i\) must also be a zero. This maintains the polynomial with real coefficients.

• The factors representing \(i\) and \(-i\) are \((x + i)\) and \((x - i)\). These factors multiply to form \(x^2 + 1\), simplifying the polynomial expression as they eliminate the imaginary components.
• Introducing a complex number without its conjugate would result in non-real (imaginary) coefficients, which contradicts the need for real coefficients.

Managing complex zeros through conjugate pairs not only preserves real integer coefficients but also ensures mathematical correctness and practicality in real-world applications.