**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 it as x = 2 and x = 3.

To convert these as factors, we have to write it as (x-2) and (x-3). The product of those factors will give the polynomial. Since we have two factors, we will get a quadratic polynomial.

Number of factors = highest order of the polynomial

Note :

Roots and zeroes are same.

**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. So, ****(x-0) (x+4) (x-5) are the factors of the required polynomial.**

**Step 3 :**

**Number of factors = 3, so we will get a cubic polynomial. ****By multiplying the above factors we will get the required cubic polynomial.**

**(x-0) (x+4) (x-5) = (x-0) (x**² - 5x + 4x - 20)

**= x (x**² - x - 20)

**= x**³ - x² - 20x

Hence the required cubic polynomial is ** x**³ - 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. **

**So, ****(x-3)(x-4/5)(x-5/2) are the factors of the required polynomial.**

**Step 3 :**

**Number of factors = 3, so we will get a cubic polynomial. By multiplying the above factors we will get the required cubic polynomial.**

= x³ - 33x²/10 + 2x - 3x² + 99x/10 - 6

= x³ - [(33/10) - 3]x² + [(2+ (99/10)]x - 6

= x³ - [(33 + 30)/10]x² + [(20+ 99)/10]x - 6

= x³ - (63/10)x² + (119/10)x - 6

= 10x³ - 63x² + 119x - 60

**Example 2 :**

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. Since 2i is the complex number, its conjugate must be the another root. ****So the required polynomial is having four roots.**

**Step 2 :**

**Now convert the values as factors.**

**So, ****(x+5)(x-0)(x-2i)(x+2i) are the factors of the required polynomial.**

**Step 3 :**

**Number of factors = 4, so we will get a polynomial of degree 4. By multiplying the above factors we will get the required polynomial.**

**(x+5)(x-0)(x-2i)(x+2i) = x (****x+5)(x-2i)(x+2i)**

**= (x**² + 5x)(x² - (2i)²)

**= (x**² + 5x)(x² - 4i²)

**= (x**² + 5x)(x² - 4(-1))

**= (x**² + 5x)(x² + 4)

**= x**⁴ + 4x² + 5x³ + 20x

Hence the required polynomial is **x**⁴ + 4x² + 5x³ + 20x

After having gone through the stuff given above, we hope that the students would have understood "Write a polynomial from its roots".

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