DEPRESSED POLYNOMIAL

Depressed Polynomial

Definition Of Depressed Polynomial

A Depressed Polynomial is the quotient that we get when a polynomial is divided by one of its binomial factors.

Example of Depressed Polynomial
In the example below, the quadratic is divided by one of its factor x - 1. Here, the quotient is the depressed polynomial.

Video Examples: Depressed Polynomials and Factor Theorem

Solved Example on Depressed Polynomial

Ques: Which of the following can be divided by the binomial factor (x - 1) to give a depressed polynomial (x - 1)?

Choices:

A. x² - 2x + 1
B. x² - 2x - 2
C. x² - 3x - 3
D. x² - 2
Correct Answer: A

Solution:

Step 1: Here, only x² - 2x + 1 can be divided by (x - 1) to give the depressed polynomial (x - 1).

Quick Summary

A depressed polynomial results from dividing a polynomial by one of its factors.
The degree of the depressed polynomial is one less than the original polynomial.
Depressed polynomials can be used to find remaining roots of a polynomial.

\[ Let P(x) be a polynomial and (x-a) be a factor of P(x). Then, \frac{P(x)}{x-a} = Q(x), where Q(x) is the depressed polynomial. \]

🍎 Teacher Insights

Emphasize the connection between factors, roots, and the depressed polynomial. Use visual aids like polynomial long division to illustrate the process. Provide ample practice problems to solidify understanding.

🎓 Prerequisites

Polynomial Division
Factoring Polynomials
Binomial Factors

Check Your Knowledge

Q1: Which of the following is the depressed polynomial of x^2 - 2x + 1 when divided by (x-1)?

x - 1 x + 1 x^2 + 1 x

Q2: What is the degree of the depressed polynomial if the original polynomial is of degree 3?

1 2 3 4

Frequently Asked Questions

Q: How do I find a depressed polynomial?
A: Divide the original polynomial by one of its known binomial factors using polynomial long division or synthetic division. The quotient is the depressed polynomial.

Q: What is the degree of a depressed polynomial compared to the original polynomial?
A: The degree of the depressed polynomial is always one less than the degree of the original polynomial.

Q: Why are depressed polynomials useful?
A: They simplify the process of finding the remaining roots of a polynomial, especially when one or more roots are already known.