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Completely factor this polynomial.

 

P(x) = x4 + 4x3 - 3x2 - 14x - 8

 Apr 6, 2022
 #1
avatar+23200 
+1

(x + 1)(x + 1)(x + 4)(x - 2)

 Apr 6, 2022
 #2
avatar+124594 
+1

Thx, geno !!!

 

Note that we can write this as :

 

x^4 + 4x^3 - (3x^2 + 14x + 8)

 

x^3 ( x + 4) -  (3x + 2) (x + 4)     simplify as

 

(x + 4) ( x^3 - 3x  - 2 )

 

Looking at the second factor, 2 is a root....so (x - 2) is also a factor

 

So....using  sythetic division to reduce the second polynomial to a quadratic, we have

 

 

2 [   1   0    -3      - 2   ]

            2     4        2

   _________________

       1   2     1        0

 

The remaining polynomial  is   x^2 + 2x + 1    which factors as  (x + 1)^2

 

So.....the factorization is

 

(x + 4) ( x - 2) ( x + 1)^2

 

cool cool cool

 Apr 6, 2022

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