Write the polynomial as the product of factors that are irreducible over the rationals. (Hint: One factor is x2 + 4.)

Question

There are three parts to this question I have the correct answer for the first part of the problem. I have tried to figure out the next two steps but keep coming up with the wrong answer. Any help on these two steps would be much appreciated.

f(x)= x⁴ − 3x³ − 3x² − 12x − 28

(B) Write the polynomial as the product of linear and quadratic factors that are irreducible over the reals. 

f(x)=?

(C) Write the polynomial in completely factored form. 

f(x)=?

Answer 1

You have to do long division:

x^2 -3x-7

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x^2 + 4 | x^4 - 3x^3 - 3x^2 - 12x - 28

x^4 0x^3 + 4x^2

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-3x^3 - 7x^2 - 12x

-3x^3 -0x^2 - 12x

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-7x^2 - 28

-7x^2 - 28

So it factors as (x^2 + 4)(x^2 - 3x - 7)

Note that for the factor x^2 - 3x - 7

the discriminant b^2 - 4ac = (-3)^2 - 4(1)(-7) = 9 - 4*-7 = 9 - -28 = 9 + 28 = 37

which is NOT a perfect square. So it cannot be factored.