Question

Find the complete factorization of P(x) = x^4-2x^3+5x^2-8x+4? the roots i found so far are :
(x+2)(x-1)(x+4)  when you multipy it out, it is x^3+5x^2+2x-8. What is the last root is what i'm trying to figure out.

Accepted Answer

If you have already found a factor of a polynomial, you can use synthetic division to find a polynomial that has smaller degree and shares the remaining factors of the polynomial.  In this case, let's use the factor x-1.
Then synthetic division gives:
1 | 1  -2  5  -8  4
  |     1   -1  4   -4
  ------------------
    1  -1   4  -4   0
This gives the lower degree polynomial:
x^3-x^2+4x-4   now this can be factored by grouping:
=x^2(x-1)+4(x-1)   apply common factoring
=(x-1)(x^2+4) 
which means the complete factorization of P(x) is P(x)=(x-1)^2(x^2+4) .
Notice that the other factors you have were not actually factors of the original polynomial.  This can be verified, since P(2)=16-2(8)+5(4)-8(2)+4 ne0 and P(4)=256-2(64)+5(16)-8(4)+4ne0 .
The complete factorization of the polynomial is P(x)=(x-1)^2(x^2+4) .

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Find the complete factorization of P(x) = x^4-2x^3+5x^2-8x+4? the roots i found so far are : (x+2)(x-1)(x+4) when you multipy it out, it is x^3+5x^2+2x-8. What is the last root is what i'm trying...

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