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.

bx3 + 70x3  =  0

(b + 70)x3  =  0

Divide each side by x3.

b + 70  =  0

b  =  -70

ax4 - 49x4  =  0

(a - 49)x4  =  0

Divide each side by x4.

a - 49  =  0 

a  =  49

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. 

-bx3 - 12x3  =  0

(b + 12)(-x3)  =  0

Divide each side by (-x3).

b + 12  =  0

b  =  -12

ax4 - 9x4  =  0

(a - 9)x4  =  0

Divide each side by x4.

a - 9  =  0 

a  =  9

Therefore, 

a  =  9

b  =  -12

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