Question

How do you find the square root of `x^4 - 4x^3 + 10x^2 - 12x +9` using the division method?

Answer 1

The answer is `=x^2-2x+3`

Explanation:

Let `y=x^2+bx+c`

Then,

`x^4-4x^3+10x^2-12x+9=(x^2+bx+c)^2`

Comparing the coefficients `c=3`

Therefore,

`x^4-4x^3+10x^2-12x+9=(x^2+bx+3)^2`

`=x^4+2(bx+3)x^2+(bx+3)^2`

`=x^4+2bx^3+6x^2+b^2x^2+6bx+9`

`=x^4+2bx^3+(6+b^2)x^2+6bx+9`

Comparing the coefficients,

`2b=-4`, `=>`, `b=-2`

The square root is `=x^2-2x+3`

LONG DIVISION METHOD

`color(white)(aaaaaaaaaaaaaaaaaa)` `x^2-2x+3`

`color(white)(aaaaaaaaaaaaaaaaaa)`##_________

`color(white)(aaaaaaaaaa)` `x^2` `color(white)(aaaaa)` `|` `x^4-4x^3+10x^2-12x+9`

`color(white)(aaaaaaaaaaaaaaaaa)` `|` `x^4`

`color(white)(aaaaaaaaaaaaaaaaa)` `|`_________

`color(white)(aaaaaa)` `2x^2-2x` `color(white)(aaaa)` `|` `color(white)(aa)` `-4x^3+10x^2-12x+9`

`color(white)(aaaaaaaaaaaa)` ###color(white)(aaaaa)#`|` `color(white)(aaa)` `-4x^3+4x^2`

`color(white)(aaaaaaaaaaaaaaaaa)` `|`_________

`color(white)(aaaa)` `2x^2-4x+3` `color(white)(aa)` `|` `color(white)(aaaaaaaaaaa)` `6x^2-12x+9`

`color(white)(aaaaaaaaaaaaaaa)` ###color(white)(aa)#`|` `color(white)(aaaaaaaaaaa)` `6x^2-12x+9`