6. In each of the following, divide

Question

f x
by
dx
, obtaining quotient
qx
and remainder
rx.

Then, write a division statement, i.e. express your answers in the form

f x  dxqx rx.

a)
 3 2  2
3 2
x  x  x   x 

b)
2 4 3 5  3
3 2
x  x  x   x 

c)
3 4  4
2
x   x  d)

4 6 6 9 2 3
3 2
x  x  x   x 

e)
3 7 5 1 3 1
3 2
x  x  x   x 

f)
 2 3 8  4
4 3 2
x  x  x  x   x 

g)
 1  1

Answer 1

a)

Dividing (x^3 + 2x^2 + x - 2) by (3x^2 - x + 2):

q(x) = (1/3)x - (7/9)
r(x) = -(2/9)x + (16/9)

Division statement: (x^3 + 2x^2 + x - 2) = (3x^2 - x + 2)(1/3)x - (7/9) + ( -(2/9)x + (16/9) )

b)

Dividing (2x^4 + 3x^3 - x^2 - 5) by (3x^2 - x - 1):

q(x) = (2/3)x^2 + (1/3)x + (1/9)
r(x) = -(8/9)x + (4/9)

Division statement: (2x^4 + 3x^3 - x^2 - 5) = (3x^2 - x - 1)(2/3)x^2 + (1/3)x + (1/9) + ( -(8/9)x + (4/9) )

c)

Dividing (3x^4 - 2x^2 + 4) by (4x^2):

q(x) = (3/4)x^2 - (2/16)
r(x) = 0

Division statement: (3x^4 - 2x^2 + 4) = (4x^2)(3/4)x^2 - (2/16)

d)

Dividing (4x^3 + 6x^2 - 6x - 9) by (2x - 3):

q(x) = 2x^2 + 6
r(x) = 3

Division statement: (4x^3 + 6x^2 - 6x - 9) = (2x - 3)(2x^2 + 6) + 3

e)

Dividing (3x^3 + 7x^2 + 5x + 1) by (3x + 1):

q(x) = x^2 + 2
r(x) = -1

Division statement: (3x^3 + 7x^2 + 5x + 1) = (3x + 1)(x^2 + 2) - 1

f)

Dividing (2x^3 + 3x^2 - 8) by (4x^3 - x^2 + x - 2):

q(x) = (1/4)
r(x) = 0

Division statement: (2x^3 + 3x^2 - 8) = (4x^3 - x^2 + x - 2)(1/4)

g)

Dividing (x - 1) by (x - 1):

q(x) = 1
r(x) = 0

Division statement: (x - 1) = (x - 1)(1) + 0