Review problems for Midterm III    Answers

Factoring polynomials with real coefficients

1.  x3 + 1 = (x + 1)( x2 – x + 1)

2.  x3 – 1 = (x - 1)( x2 + x + 1)

3.  x4 – 1 = (x – 1)(x + 1)( x2 + 1)

4.  x4 + 1 = (x2 + 21/2x + 1)( x2 - 21/2 x + 1)

5.  x4 + 4 = (x2 + 2x + 2)( x2 - 2x + 2)

6.  x4 – 16 = (x – 2)(x + 2)( x2 + 4)

7.  x6 – 1 = (x + 1)( x2 – x + 1) (x - 1)( x2 + x + 1)

8.  x6 + 1 =  (x2 + 1)( x2 + 31/2 x + 1) (x2 - 31/2x + 1)

9.   x3 – 2x2 + 4x – 8 = (x – 2)( x2 + 4)

10.  x4+ 10x2 + 24 = ( x2 + 4) ( x2 + 6)

11.  x4 – 10x2 + 24  = (x + 2)(x – 2)(x + 61/2 )(x -  61/2 )

Simple Harmonic Motion

p.323: #2   x = A sin(at + b)

v = aA cos(at + b).

The equation is now easily checked by direct substitution.

#4   x = A sin(at + b)  with  a  not zero.

0 = x(0) =  A sin(b).  So  sin(b) = 0.  Since you only need one possible

formula for  x(t), select  b = 0 for simplicity,

Then,  x(1) = 0 = A sin(a).  So  sin(a) = 0.  Select  a = pi for simplicity.

Then v(t) = pi*A cos(pi*t), and  -3 = v(0) = pi*A.

Therefore  A = -3/pi

Hyperbolic functions

p. 329: #3   cosh(x + y) = (1/2)*[ex+y + ex-y]

coshx cosh y + sinh x sinh y =

(1/2)*[ex + e-x] (1/2)*[ey + e-y] +  (1/2)*[ex - e-x] (1/2)*[ey - e-y].

Simplify this long expression, and it will be equal to the shorter one.

#9   2 sinh2 x =  2*(1/4)[ [ex - e-x]2 =  (1/2)[ [e2x - 2 + e-2x]

= (1/2)[ [e2x  + e-2x] – 1 = cosh 2x – 1.