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Let a and b real numbers such that x^4+2x^3-4x^2+ax+b = (Q(x))^2 for some polynomial Q(x). What is the value of a + b?

 Oct 21, 2022
 #1
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Q(x)=(lx^2+mx+n)

Q(x)^2=(l^2x^2 + 2mlx^3 + (2ln+m^2)x^2+2mnx+n^2)

Therefore

l^2=1

2ml=2

ml=1

2ln+m^2=-4

a=2mn

b=n^2

 

If l is one, so is m.

If l is -1, so is m.

 

l has to be negative, because of 2ln+m^2=-4, which means 2ln is negative.

2ln=-5

n=2.5

b=6.25

a=-5

a+b=1.25

(I think)

 Oct 21, 2022
edited by WhyamIdoingthis  Oct 21, 2022

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