CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 8. Systems and Matrices

Notice the squared term, linear term and constant term form a system of three equations with three variables.

A+B= 7

C= 1

3 A= 6

In this case it is easy to see thatA= 2 ,B= 5 ,C=1. Often, the resulting system of equations is more complex and
would benefit from your knowledge of solving systems using matrices.

(^7) xx(^2 x+ (^2) +x+ 3 ) (^6) =x (^2) + (^5) x 2 x++ 31
Example B
Decompose the following rational expression.
5 x^4 − (^3) (xx−^3 − 1 )x (^32) x+ 24 x− 1
Solution: First identify the denominators of the partial fractions.
5 x^4 − 3 x^3 −x^2 + 4 x− 1
(x− 1 )^3 x^2 =

A

x− 1 +

B

(x− 1 )^2 +

C

(x− 1 )^3 +

D

x+

E

x^2

When the entire fraction is multiplied through by(x− 1 )^3 x^2 the equation results to:

5 x^4 − 3 x^3 −x^2 + 4 x− 1

=A(x− 1 )^2 x^2 +B(x− 1 )x^2 +Cx^2 +D(x− 1 )^3 x+E(x− 1 )^3

Multiplication of each term can be done separately to be extra careful.

Ax^4 − 2 Ax^3 +Ax^2

Bx^3 −Bx^2

Cx^2

Dx^4 − 3 Dx^3 + 3 Dx^2 −Dx

Ex^3 − 3 Ex^2 + 3 Ex−E

Group terms with the same power ofxand set equal to the corresponding term.

5 x^4 =Ax^4 +Dx^4

− 3 x^3 =− 2 Ax^3 +Bx^3 − 3 D^3 +Ex^3

−x^2 =Ax^2 −Bx^2 +Cx^2 + 3 Dx^2 − 3 Ex^2

4 x=−Dx+ 3 Ex

− 1 =−E

From these 5 equations, everyxcan be divided out. Assume thatx 6 =0 because if it were, then the original expression
would be undefined.