Question

The order of the numerator is lower than the order of the denominator. If this isn't the case, it can be achieved through polynomial division, where the N(x)/D(x) becomes something in the form α + rβ = [β is remainder] α + β/D(x). Now decompose β/D(x) as usual.

a) Perform polynomial division
b) Simplify the expression
c) Evaluate the expression for a specific value of x
d) Find the value of α in the expression

Answer 1

Polynomial division is used to divide
`N(x) by D(x)` when the numerator's order is higher than the denominator's, yielding a quotient
`(α)`and a remainder
`(β)`. Simplify by factoring and reduce, evaluate by substituting x, and α is found during the division process.

Step-by-step explanation:

When working with rational expressions where the order of the numerator is higher than the order of the denominator, polynomial division can be employed to rewrite the expression. This process transforms the initial fraction
`N(x)/D(x)` into a form where α represents the quotient andis the remainder resulting in a new expression
`α + β/D(x).`

In order to perform polynomial division:

1. Divide the leading term of the numerator by the leading term of the denominator.
2. Subtract the obtained polynomial from the numerator.
3. Repeat the process until the order of the numerator is less than the order of the denominator.

Simplifying the expression involves factoring and reducing common terms. To evaluate the expression for a specific value of x simply substitute x with the given value in both α and
`β` and perform the arithmetic operations. The value of α the quotient without the remainder, is determined aft
`β`er the division process and before inclusion of the remainder term (β/D(x)).