FINDING SQUARE ROOT OF A POLYNOMIAL

In this section, you will learn how to find square root of a polynomial using long division. 

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Note : 

Before proceeding to find the square root of a polynomial, one has to ensure that the degrees of the variables are in descending or ascending order.

Examples

Example 1 :

Find the square root of the following polynomial :

x⁴ - 4x³ + 10x² - 12x + 9

Solution :

Therefore the square root of the given polynomial is

|x² - 2x + 3|

Example 2 :

Find the square root of the following polynomial :

4x⁴ + 8x³ + 8x² + 4x + 1

Solution :

Therefore the square root of the given polynomial is

|2x² + 2x + 1|

Example 3 :

Find the square root of the following polynomial :

9x⁴ - 6x³ + 7x² - 2x + 1

Solution :

Therefore the square root of the given polynomial is

|3x² - x + 1|

Example 4 :

Find the square root of the following polynomial :

4 + 25x² - 12x - 24x³ + 16x⁴

Solution :

First arrange the term of the polynomial from highest exponent to lowest exponent and find the square root. 

Then, 

16x⁴ - 24x³ + 25x² - 12x + 4

Therefore the square root of the given polynomial is

|4x² - 3x + 2|

Example 5 :

Find the values of a and b if the following polynomial is a perfect square

4x⁴ - 12x³ + 37x² + ax + b

Solution :

Because the given polynomial is a perfect square,

a + 42  =  0  and  b - 49  =  0

Solving the above equations for a and b, we get

a  =  -42

b  =  49

Example 6 :

Find the values of a and b if the following polynomial is a perfect square

x⁴ - 4x³ + 10x² - ax + b

Solution :

Because the given polynomial is a perfect square,

-a + 12  =  0  and  b - 9  =  0

Solving the above equations for a and b, we get

a  =  12

b  =  9

Example 7 :

Find the values of a and b if the following polynomial is a perfect square

ax⁴ + bx³ + 109x² - 60x + 36

Solution :

Here a and b are being the coefficients of x⁴ and x³ respectively. 

To solve for a and b, always they have to come at last. 

So, write the given polynomial from lowest exponent to highest exponent. 

36 - 60x + 109x² + bx³ + ax⁴

Because the given polynomial is a perfect square,

bx³ + 70x³  =  0  and  ax⁴ - 49x⁴  =  0

Solve the above equations for a and b.

Therefore, 

a  =  49

b  =  -70

Example 8 :

Find the values of a and b if the following polynomial is a perfect square

ax⁴ - bx³ + 40x² + 24x + 36

Solution :

Here a and b are being the coefficients of x⁴ and x³ respectively. 

To solve for a and b, always they have to come at last. 

So, write the given polynomial from lowest exponent to highest exponent. 

36 + 24x + 40x² - bx³ + ax⁴

Because the given polynomial is a perfect square,

-bx³ - 12x³  =  0  and  ax⁴ - 9x⁴  =  0

Solve the above equations for a and b. 

Therefore, 

a  =  9

b  =  -12

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