|
The square root of a polynomial $f$ denoted by $\sqrt{f}$ is any polynomial $g$ having the square $g^2$ equal to $f$ For example, $\sqrt{9x^2\!-\!30x\!+\!25} = 3x\!-\!5$ or $-3x\!+\!5$
A polynomial needs not have a square root, but if it has a square root $g$ then also the opposite polynomial $-g$ is its square root.
Algorithm. The idea of the squaring formula $$(a\!+\!b\!+\!c+..)^2 =(a)a+(2a\!+\!b)b+(2a\!+\!2b\!+\!c)c+..$$ (see the square of sum) gives a method for getting the square root of a polynomial:
- The terms of the radicand are ordered according to the rising or falling powers of certain letter (the first term must have a positive coefficient and even exponents).
- The leading term of the root is equal to the square root of the first term of the radicand.
- The second term of the root is equal to the first term of the first remainder divided by the double leading term.
- The third term of the root is equal to first term of the second remainder divided by the double leading term.
- And so on.
In the examples below, on the left under the lines there are the remainders, on the right under the lines the corresponding sums.
Example 1. $\sqrt{9x^4\!+\!6x^3\!-\!11x^2\!-\!4x\!+\!4} =$ ?
| $\sqrt{} $ $(9x^4$ $+6x^3$ $-11x^2$ $-4x$ $+4)$ $=$ $\pm$ $(3x^2$ $ +x$ $-2)$ $ $ $ 9x^4$ $ $ $ $ $ $ $ $ $ $ $ $ $ 3x^2$ $ $ $ $ $ $ $ $ $ 6x^3$ $-11x^2$ $ $ $ $ $ $ $
$ $ 6x^2$ $ +x$ $ $ $ $ $ $ $ 6x^3$ $ +x^2$ $ $ $ $ $ $ $ $ $ $ $ x$ $ $ $ $ $ $ $ $ $-12x^2$ $-4x$ $ +4$ $ $ $ $ $ 6x^2$ $ +2x$ $-2$ $ $ $ $ $ $ $-12x^2$ $-4x$ $ +4$ $ $ $ $ $ $ $ $ $-2$ $ $ $ $ $ $ $ $ $ $ $ 0$ $ $ $ $ $ $ $ $ $ $ |
|
|
|
|
|
|
|
|
|
|
|
|
Example 2. $\sqrt{x^6\!-\!2x^5\!-\!x^4\!+\!3x^2\!+\!2x\!+\!1} =$ ?
| $\sqrt{}$ $(1$ $+2x$ $+3x^2$ $$&$-x^4$&$-2x^5$&$+x^6)$&$=$& $\pm$&$(1$&$ +x$&$+x^2$&$-x^3)$\\ $$ |
$1$ $$&$$ |
$$&$$ |
$$&$$ |
$$&$$ |
$1$ $$&$$ |
$$\\ \cline{2-2} \cline{11-11} $$ |
$$&$2x$&$+3x^2$&$$ |
$$&$$ |
$$&$$ |
$$&$2$&$+x$&$$ |
$$\\ $$ |
$$&$2x$&$+x^2$&$$ |
$$&$$ |
$$&$$ |
$$&$$ |
| $$& $$ |
$$&$2x^2$&$$ |
$-x^4$ $$&$$ |
$$&$$ |
$2$ $+2x$ $+x^2$ $$\\ $$ |
$$&$$ |
$2x^2$ $+2x^3$ $+x^4$ $$&$$ |
$$&$$ $$&&$$ |
$x^2$ $$\\ \cline{4-6} \cline{11-13} $$ |
$$&$$ |
$$&$-2x^3$&$-2x^4$&$-2x^5$&$+x^6$&$$ |
$$&$2$&$+2x$&$+2x^2$&$-x^3$\\ $$ |
$$&$$ |
$$&$-2x^3$&$-2x^4$&$-2x^5$&$+x^6$&$$ |
$$&$$ |
$$&$$ |
| 
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)