Practice Exercises for Lecture 4:
Factor the following polynomials.
P(x) = x2โ6x+ 8
P(x)=4x2โ16x+ 15
P(x) = x3โ5x2+ 6x
P(x) = x3โ6x+ 5
Hint: In the last case note that x= 1 is a zero (root) of x3โ6x+ 5. Use synthetic
division to write x3โ6x+ 5 = (xโ1)(x2+ax +b).
Find a partial fractions expansion for the following rational functions. The general form is
given to you
7xโ2
x2โ6x+ 8 =a
xโ2+b
xโ4
1
4x2โ16x+ 15 =a
(xโ3/2) +b
(xโ5/2)
x2+x+ 1
x3โ3x2+ 2x=a
x+b
(xโ1) +c
(xโ2)
x2โx+ 7
x3(xโ1) dx =a
x+b
x2+c
x3+d
xโ1
Finish calculating the following integral, which was reduced using synthetic division in lecture
Zx4โ5x2
x3โ3x2+ 2xdx
Give the correct form of the partial fractions expansion for the following rational functions.
Indicate if you need to do synthetic division and, if so, what the form of the result will be.
You neednโt actually calculate the relevant coefficients.
x2
โ8
x3(xโ1)(xโ2)
x5
โ7x3
x2(x2โ2x+1)
x3
โ1
x(x2โ3x+2)