find the remaining zeros of f

Question

f(x)=x⁴+8x³+2x²-80x-75 given that -5 is a zero of multiplicity 2

Answer 1

Since we already know a zero, we can easily use synthetic division.  This division process divides the coefficients of f(x) by the zero.  When we divide, the last digit in the quotient must be zero.

 

 

-5 | 1      8        2        -80      -75

 

             -5      -15        65       75

_____________________________

 

      1      3      -13       -15         0

 

 

 

The digits under the line is the quotient.  As you can see, the last digit is 0.  So we can say that the factored form is

 

(x + 5)(x³ + 3x² - 13x - 15) = 0

 

 

Since the zero has a multiplicity of 2, we can divide the coefficients of the larger factor by -5.

 

 

 

-5 | 1       3        -13        -15

 

              -5          10         15

___________________________

  

      1      -2          -3           0

 

 

 

So now, we have

 

(x + 5)(x + 5)(x² - 2x - 3) = 0

 

 

Factoring completely, we get

 

(x + 5)(x + 5)(x - 3)(x + 1) = 0

 

 

So the remaining zeros are

 

x = -1

x = 3