# 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.

lfryerda | Certified Educator

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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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