Expert: Sherman D. - 1/23/2005

Question
Im confused about a few problems, theyre similar but I do not understand them.

1. write f(x)=x^5+4x^4-2x^3-14^2-3x-18 as the product of linear factors

2.Descartes rule-- determine the possible number of positive and negative real zeros of:        (f)x=4x^3+8x^2-3x+9

3. Find the REAL zeros of the function:        (f)x =x^3-9x^2+26x-21

4.Descartes: Determine number of positive and negative zeros of the following functions:       A.f(x) =-8x^3 2x^2+7x

B. f(x) =-7x^2+3x-5

5. Rational Zero test list all rational zeros of:
  f(x)=-9x^3=10x^2-6x-2

6. find a polynomial with integer coefficients that has the zeros -1, 7i, and -7i
      Choices:
   x^3+x^2+49x+49
   x^3-x^2+49x-49
   x^3+x^2-49x-49
   or none of these  

Answer
1. write f(x)=x^5+4x^4-2x^3-14^2-3x-18 as the product of linear factors
if by this you mean
x^5+4x^4-2x^3-14x^2-3x-18
then the factors are
(x + 3)(x + 3)(x + i)(x - i)(x - 2)
Go to www.quickmath.com, click on Factor, under Algebra, then type in your problem like this
x^5 + 4x^4 - 2x^3 - 14x^2 - 3x - 18

When you do it like that, it will give you
(x - 2)(x + 3)^2(x^2 + 1)
but if you already know about complex/imaginary values, you can go ahead and type it out like this
(x - 2)(x + 3)(x + 3)(x - i)(x + i)

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2.Descartes rule-- determine the possible number of positive and negative real zeros of:      
(f)x = 4x^3 + 8x^2 - 3x + 9
www.quickmath.com, click on Solver located under Equations, then type in your problem like this
4x^3 + 8x^2 - 3x + 9 = 0

-----------------------------------------------------------

3. Find the REAL zeros of the function:      
(f)x =x^3-9x^2+26x-21
www.quickmath.com, click on Solver located under Equations, then type in your problem like this
x^3 - 9x^2 + 26x - 21 = 0

-----------------------------------------------------------

4.)
Never used Descartes
Descartes: Determine number of positive and negative zeros of the following functions:    
A.)f(x) = -8x^3 2x^2 + 7x
factors to
-x(8x^2 2x - 7)
sorry i don't know what sign you meant in between -8 and 2.

B.) f(x) = -7x^2 + 3x - 5
x = (-3 ± sqrt(3^2 - 4(-7)(-5)))/2(-7)
x = (-3 ± sqrt(9 - 140))/-14
x = (-3 ± sqrt(131))/-14
x = (3/14) ± (-1/14)sqrt(131)

For both, i would use

x = (-b ± sqrt(b^2 - 4ac))/2a

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5.) If by this you mean
Rational Zero test list all rational zeros of:
f(x) = -9x^3+10x^2-6x-2
www.quickmath.com, click on Solver located under Equations, then type in your problem like this
-9x^3 + 10x^2 - 6x - 2 = 0
-----------------------------------------------------------

6.) find a polynomial with integer coefficients that has the zeros -1, 7i, and -7i
   Choices:
  x^3+x^2+49x+49
  x^3-x^2+49x-49
  x^3+x^2-49x-49
  or none of these

the zeros become

(x + 1)(x - 7i)(x + 7i)
(x + 1)(x^2 + 7ix - 7ix - 49i^2)
(x + 1)(x^2 - 49(-1))
(x + 1)(x^2 + 49)
x^3 + 49x + x^2 + 49
x^3 + x^2 + 49x + 49

The answer is the first equation you had.

Sorry i couldn't be of much help. I'm just not good with all cube problems.