Writing Polynomial Equations Given Roots Calculator
WRITE A POLYNOMIAL FROM ITS ROOTS
Root is nothing but the value of the variable that we find in the equation.To get a equation from its roots, first we have to convert the roots as factors. By multiplying those factors we will get the required polynomial.
For example,
2 and 3 are the roots of the polynomial then we have to write them as
x = 2 and x = 3
To convert these as factors, we have to write them as
(x - 2) and (x - 3)
The product of those factors will give the polynomial. Because we have two factors, we will get a quadratic polynomial.
Number of factors = Highest exponent of the polynomial
Note :
Roots and zeroes are same.
Examples
Example 1 :
Write the polynomial function of the least degree with integral coefficients that has the given roots.
0, -4 and 5
Solution :
Step 1 :
0, -4 and 5 are the values of x.
So we can write these values as
x = 0, x = -4 and x = 5
Step 2 :
Now convert the values as factors.
(x - 0), (x + 4), (x - 5) are the factors of the required polynomial.
Step 3 :
Number of factors = 3
Then, we will get a cubic polynomial.
By multiplying the above factors we will get the required cubic polynomial.
So, the required polynomial is
= (x - 0)(x + 4)(x - 5)
= (x-0)(x2 - 5x + 4x - 20)
= x(x2 - x - 20)
= x3 - x2 - 20x
Example 2 :
Write the polynomial function of the least degree with integral coefficients that has the given roots.
3, 4/5 and 5/2
Solution :
Step 1 :
3, 4/5 and 5/2 are the values of x. So we can write these values as
x = 3, x = 4/5 and x = 5/2
Step 2 :
Now convert the values as factors.
(x - 3), (x - 4/5) and (x - 5/2) are the factors of the required polynomial.
Step 3 :
Number of factors = 3
Then, we will get a cubic polynomial.
By multiplying the above factors we will get the required cubic polynomial.
So, the required polynomial is
= (x - 3)(x - 4/5)(x - 5/2)
= (x - 3)(x2 - 5x/2 - 4x/5 + 2)
= (x - 3)(x2 - 33x/10 + 2)
= (x - 3)(x2 - 33x/10 + 2)
= x3 - 33x2/10 + 2x - 3x2+ 99x/10 - 6
Combine the like terms.
= x3 - 63x2/10 + 119x/10 - 6
Example 3 :
Write the polynomial function of the least degree with integral coefficients that has the given roots.
-5, 0 and 2i
Solution :
Step 1 :
-5, 0 and 2i are the values of x.
Because 2i is the complex number, its conjugate must also be another root.
So, the required polynomial is having four roots.
Step 2 :
Now convert the values as factors.
(x + 5), (x - 0), (x - 2i) and (x + 2i) are the factors of the required polynomial.
Step 3 :
Number of factors = 4
Then, we will get a polynomial of degree 4 .
By multiplying the above factors we will get the required polynomial.
So, the required polynomial is
= (x + 5)(x - 0)(x - 2i)(x + 2i)
= x( x + 5)(x - 2i)(x + 2i)
= (x2 + 5x)[x2 - (2i)2 ]
= (x2 + 5x)(x2 - 4i2)
= (x2 + 5x)[(x2 - 4(-1)]
= (x2 + 5x)[(x2 - 4(-1)]
= (x2 + 5x)(x2 + 4)
= x4 + 4x2 + 5x3 + 20x
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Writing Polynomial Equations Given Roots Calculator
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