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.
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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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