Write a third-degree polynomial equation with real coefficients and the given roots. -2,3-i

Name: Problem 28, Chapter 5: College Algebra
Uploaded: 2020-07-06T21:53:45-08:00
Duration: 2 min 3 s
Channel: James Kiss
Description: Write a third-degree polynomial equation with real coefficients and the given roots. -2,3-i

step 1 We want to write a third degree polynomial that has the roots negative 2, 3 minus I. And then be

Write a third-degree polynomial equation with real...

Key Concepts

Real Coefficient Polynomial

When constructing a polynomial with real coefficients, it is essential to ensure that the imaginary parts cancel out through the inclusion of complex conjugate pairs. This is why, in the given problem, a complex root must be accompanied by its conjugate, so that the product of the associated factors yields a quadratic factor with real coefficients that contributes to the overall polynomial with real coefficients.

Degree of Polynomial

The degree of a polynomial is determined by the highest power of x in its expanded form. In this case, a third-degree polynomial means the polynomial will have three factors (or roots) when all multiplicities are taken into account. This concept is directly linked to the fundamental theorem of algebra, which states that a non-zero, degree-n polynomial has exactly n roots (counting multiplicities).

Complex Conjugate Root Theorem

For any polynomial with real coefficients, if a complex number is a root, then its complex conjugate is also a root. This means that if one root is 3 - i, then 3 + i must also be a root. This ensures that when factors involving these complex numbers are multiplied, the resulting expressions combine to produce real coefficients.

Polynomial Construction from Roots

A polynomial can be constructed by taking factors corresponding to each of its roots. Each root r contributes a factor of the form (x - r). Multiplying these factors together gives a polynomial that has exactly the given roots. For a third-degree polynomial, three factors are multiplied together, reflecting the three roots (with the complex roots counted as a conjugate pair along with the additional given real root).

Transcript

Please give Ace some feedback