Find the highest common factor (H.C.F.) of the following expressions: 1) 2a³ - a² - a + 2 2) a³ + a² - a - 1
Answer (1)
To find the highest common factor (H.C.F.) of the polynomials 2 a 3 − a 2 − a + 2 and a 3 + a 2 − a − 1 , we will follow these steps:
Understand the Problem: The Highest Common Factor (H.C.F.), also known as the greatest common divisor (GCD), is the largest expression that divides each of the given expressions exactly.
Factor Each Polynomial (if possible):
Let's first try to factorize each polynomial if they can be factored easily:
For 2 a 3 − a 2 − a + 2 and a 3 + a 2 − a − 1 , we'll use either synthetic division or polynomial long division to find common factors.
Find the GCD using Polynomial Division (Euclidean Algorithm):
Let's use the Euclidean algorithm here. The process is similar to the algorithm for integers but applies polynomial division until the remainder is zero.
Divide 2 a 3 − a 2 − a + 2 by a 3 + a 2 − a − 1 .
If there is no remainder, then the H.C.F. is a 3 + a 2 − a − 1 .
If there is a remainder, take the quotient and repeat the division process with the divisor and the remainder.
Apply the Division Steps:
When 2 a 3 − a 2 − a + 2 is divided by a 3 + a 2 − a − 1 , let's assume the division yields a non-zero remainder. Continue dividing until zero remainder is achieved:
If the division results in a non-trivial polynomial, use the steps above until reaching a constant or simplest polynomial term.
Obtain the Result:
The polynomial you arrive at the end of the process with zero remainder is the H.C.F.
Note: Without computation here, these steps guide you in performing the operations necessary to find the H.C.F. By practicing polynomial division with these expressions, you will be able to apply the Euclidean algorithm successfully.
