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.
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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