Square Root of Polynomials

Square root of a polynomial

Since (t + u)^2 = t^2 + 2tu + u^2, the square root of t^2 + 2tu + u^2 is \pm(t + u). A study of this last form shows how to extract the square root of any polynomial that is a perfect square.

The square root of t^2 (the first term) is t, the first term of the root. Subtracting t^2 leaves 2tu + u^2. The next term of the root, u, is found by dividing the first term of this remainder, 2tu, by 2t, twice the root already found. Then 2t + u, the complete divisor, times u gives exactly 2tu + u^2, and nothing remains.

Example 1. Extract the square root of 4x^4 - 20x^3 + 37x^2 - 30x + 9.

	Work	Root: 2x^2 - 5x + 3
	4x^4 - 20x^3 + 37x^2 - 30x + 9
(2x^2)^2 =	4x^4	first term 2x^2 = \sqrt{4x^4}
First trial divisor 2(2x^2) = 4x^2	-20x^3 + 37x^2	-20x^3 \div 4x^2 = -5x
First complete divisor 4x^2 - 5x	-20x^3 + 25x^2 = (4x^2 - 5x)(-5x)
Second trial divisor 2(2x^2 - 5x) = 4x^2 - 10x	12x^2 - 30x + 9	12x^2 \div 4x^2 = 3
Second complete divisor 4x^2 - 10x + 3	12x^2 - 30x + 9 = (4x^2 - 10x + 3)(3)	remainder 0

Therefore the square roots are \pm(2x^2 - 5x + 3).

Rule

Arrange the terms according to descending powers of some letter.
Extract the square root of the first term; write it (with plus sign only) as the first term of the root, and subtract its square from the polynomial.
Double the root already found for a trial divisor, divide the first term of the remainder by it, and write the quotient as the next term of the root.
Annex that quotient to the trial divisor to make the complete divisor; multiply the complete divisor by the new term of the root and subtract.
If terms remain, double the whole root found so far for a new trial divisor and proceed as before.
Inclose the root in a parenthesis preceded by the sign \pm.

Example 2 (fractional terms). Extract the square root of \dfrac{4x^2}{y^2} + \dfrac{12x}{y} + 7 - \dfrac{3y}{x} + \dfrac{y^2}{4x^2}.

The first term of the root is \dfrac{2x}{y}. The trial divisor \dfrac{4x}{y} goes into \dfrac{12x}{y} 3 times, and \left(\dfrac{4x}{y} + 3\right)3 = \dfrac{12x}{y} + 9 leaves -2 - \dfrac{3y}{x} + \dfrac{y^2}{4x^2}. The new trial divisor \dfrac{4x}{y} + 6 goes into -2 \;-\dfrac{y}{2x} times, and \left(\dfrac{4x}{y} + 6 - \dfrac{y}{2x}\right)\left(-\dfrac{y}{2x}\right) uses up the rest. The square roots are \pm\left(\dfrac{2x}{y} + 3 - \dfrac{y}{2x}\right).

Check: square the root; the result should be the original polynomial.

The fourth root of an expression is the square root of its square root.

Typing answers: type the root with the sign of its first term positive, like 2x^2 - 5x + 3; the \pm is understood (you may also type ±(2x^2 - 5x + 3)). Fractions: x^2 + 3x + 1/2, a/2 - 3/a.

From the history of algebra

The process of extracting the square root of numbers was familiar to mathematicians long before they knew how to find the square root of polynomials. This fits a general pattern: methods for working with letters grew out of the similar operations on numerals. Applying the rules for the square root of numbers to polynomials is generally credited to Robert Recorde (1510–1558), author of the earliest English work on algebra that we know. His book, The Whetstone of Witte (1557), gives an accurate idea of the algebra of the time and had a very wide influence. It is also the book in which the sign = first appeared.

Exercises

0 of 21 solved

How to type math

Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.

Extract the square root of:

a^{4} + 3 a^{2} + 2 a^{3} + 2 a + 1

24 x^{2} - 32 x + 16 + x^{4} - 8 x^{3}

21 c^{2} + c^{4} + 20 c - 10 c^{3} + 4

n^{6} + 9 n^{2} + 10 n^{3} + 25 - 6 n^{4} - 30 n

19 a^{2} - 11 a^{4} + 4 a^{6} - 30 a + 4 a^{5} + 14 a^{3} + 25

c^{4} - 4 c^{3} d + 6 c^{2} d^{2} - 4 c d^{3} + d^{4}

30 x y^{3} + 25 y^{4} - 11 x^{2} y^{2} - 12 x^{3} y + 4 x^{4}

- 36 a^{4} x + 36 a^{2} x^{2} + 9 a^{6} - 24 a^{3} x^{2} + 16 x^{4} + 48 a x^{3}

9 c^{4} - 2 a^{2} b^{2} c^{2} + 4 a^{3} b^{3} c + a^{4} b^{4} - 12 a b c^{3}

2 a^{2} x c^{3} - 4 x c^{3} - 4 a^{2} x^{2} + 4 x^{2} + c^{6} + a^{4} x^{2}

4 - \dfrac{4 c^{2}}{5} + \dfrac{c^{4}}{25}

\dfrac{9}{a^{2}} + \dfrac{a^{2}}{4} - 3

x^{4} - 4 x^{3} + 5 x^{2} - 2 x + 1 \cdot \dfrac{1}{4}

x^{4} + 6 x^{3} + \dfrac{29 x^{2}}{3} + 2 x + 1 \cdot \dfrac{1}{9}

4 a^{4} + \dfrac{4 a^{3}}{3} - \dfrac{35 a^{2}}{9} - \dfrac{2 a}{3} + 1

\dfrac{9}{4} - \dfrac{127 m^{2}}{18} + \dfrac{25 m^{4}}{4} - 2 m + \dfrac{10 m^{3}}{3}

9 c^{4} - 12 c^{3} + 4 c^{2} - \dfrac{4}{c} + 1 \dfrac{1}{c^{4}} + 6

\dfrac{m^{2}}{n^{2}} + \dfrac{n^{2}}{m^{2}} + \dfrac{17}{4} - \dfrac{5 m}{n} + \dfrac{5 n}{m}

\dfrac{a^{4}}{25 c^{4}} + \dfrac{2 a^{2}}{c^{3}} + \dfrac{117}{5 c^{2}} - \dfrac{40}{a^{2} c} + \dfrac{16}{a^{4}}

\dfrac{a^{2}}{4 c^{4}} + 9 + \dfrac{4 c^{2}}{25 a^{4}} - \dfrac{3 a}{c^{2}} + \dfrac{2}{5 a c} - \dfrac{12 c}{5 a^{2}}

Extract the fourth root of the expressions in Exercises 2 and 6. (The fourth root of a number is the square root of its square root.)