Question

How do you find the square root of `4 + 25x^2 - 12x-24x^3+16x^4` using the division method?

Answer 1

`sqrt(16x^4-24x^3+25x^2-12x+4) = 4x^2-3x+2`

Explanation:

Given:

`4+25x^2-12x-24x^3+16x^4`

Rearrange in standard form with descending powers of `x`:

`16x^4-24x^3+25x^2-12x+4`

Note that if this polynomial has a square factor then that will also be a factor of its derivative:

`64x^3-72x^2+50x-12 = 2(32x^3-36x^2+25x-6)`

So we want to find the GCF of these polynomials.

Multiplying the original quartic by `2`, we get:

`32x^4-48x^3+50x^2-24x+8`

`=x(32x^3-36x^2+25x-6)-12x^3+25x^2-18x+8`

Now:

`3(32x^3-36x^2+25x-6)+8(-12x^3+25x^2-18x+8)`

`= 96x^3-108x^2+75x-18-96x^3+200x^2-144x+64`

`= 92x^2-69x+46`

`= 23(4x^2-3x+2)`

Then we find:

`(4x^2-3x+2)^2 = 16x^4-24x^3+25x^2-12x+4`

as required.

So:

`sqrt(16x^4-24x^3+25x^2-12x+4) = 4x^2-3x+2`

Answer 2

`sqrt(16x^4-24x^3+25x^2-12x+4) = 4x^2-3x+2`

Explanation:

Given:

`4+25x^2-12x-24x^3+16x^4 = 16x^4-24x^3+25x^2-12x+4`

Note that if this is a square of a simpler polynomial, then it must be expressible in one of two forms:

`(4x^2+ax-2)^2`

`(4x^2+ax+2)^2`

So we have one of the following:

`(4x^2+ax-2)^2 = 16x^4+8ax^3+(a^2-16)x^2-4ax+4`

`(4x^2+ax+2)^2 = 16x^4+8ax^3+(a^2+16)x^2+4ax+4`

Equating the coefficients of `x^3`, in either case we find `8a=-24` and hence `a=-3`

Then matching the coefficients of `x`, we have `-12 = +-4a` and we find that we need `+4ax`

So:

`16x^4-24x^3+25x^2-12x+4 = (4x^2-3x+2)^2`

So:

`sqrt(16x^4-24x^3+25x^2-12x+4) = 4x^2-3x+2`