SOLUTION: Find all zeros of the function and write the polynomial as a product of linear factors. {{{ f(x) = x^4 + 6x^3 + 17x^2 + 54x + 72 }}} A. {{{ f(x) = (x - 4)(x + 2)(x - 3)(x + 3

Algebra ->  Coordinate-system -> SOLUTION: Find all zeros of the function and write the polynomial as a product of linear factors. {{{ f(x) = x^4 + 6x^3 + 17x^2 + 54x + 72 }}} A. {{{ f(x) = (x - 4)(x + 2)(x - 3)(x + 3      Log On


   

Question 472580: Find all zeros of the function and write the polynomial as a product of linear factors.
+f%28x%29+=+x%5E4+%2B+6x%5E3+%2B+17x%5E2+%2B+54x+%2B+72+
A. +f%28x%29+=+%28x+-+4%29%28x+%2B+2%29%28x+-+3%29%28x+%2B+3%29+
B. +f%28x%29+=+%28x+%2B+4%29%28x+%2B+2%29%28x+-+3i%29%28x+%2B+3i%29+
C. +f%28x%29+=+%28x+-+1%29%28x+-+8%29%28x+-+3i%29%28x+%2B+3i%29+
D. +f%28x%29+=+%28x+-+i%28sqrt8%29%29%28x+%2B+i%28sqrt8%29%29%28x+-+3%29%28x+%2B3%29+
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
+f%28x%29+=+x%5E4+%2B+6x%5E3+%2B+17x%5E2+%2B+54x+%2B+72+


If you know about DesCartes rule of signs, you don't have to do
any work.  You will notice that there are no sign changes in 
going from the first term to the last.  Therefore it can have 
no positive zeros.  Only one of the choices has no positive 
zeros, so you don't have to do any work.
Look at

B. +f%28x%29+=+%28x+%2B+4%29%28x+%2B+2%29%28x+-+3i%29%28x+%2B+3i%29+

and notice it's the only one that has no factors of the form 
x+-+A_POSITIVE_REAL_NUMBER like (x - 4) or (x - 3). 

Setting each factor of B to zero gives:

x + 4 = 0        x + 2 = 0      x - 3i = 0       x + 3i = 0 
    x = -4           x = -2          x = 3i           x = -3i

B is the only one that has only negative and imaginary zeros.
The other answers have at least one negative zero which Descartes'
rule of signs shows us. 

[Setting any of the others choices to zero will give some negative 
zeros.  For instance if you set x - 4 = 0 you get a positive zero 4,
and so you eliminate all that have a factor "x - a positive number" 

Edwin