**Problem 1 :**

Write the polynomial function of the least degree with integral coefficients that has the given roots.

0, -4 and 5

**Problem 2 :**

Write the polynomial function of the least degree with integral coefficients that has the given roots.

3, 4/5 and 5/2

**Problem 3 :**

Write the polynomial function of the least degree with integral coefficients that has the given roots.

-5, 0 and 2i

**Problem 1 :**

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)(x ^{2}** - 5x + 4x - 20)

**= x(x ^{2}** - x - 20)

**= x ^{3}** - x

**Problem 2 :**

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)(x ^{2} - 5x/2 - 4x/5 + 2)**

**= ****(x - 3)(x ^{2} - 33x/10 + 2)**

**= ****(x - 3)(x ^{2} - 33x/10 + 2)**

**= ****x ^{3} - 33x^{2}/10 + 2x - 3x^{2 }+ 99x/10 - 6**

**Combine the like terms. **

**= ****x ^{3} - 63x^{2}/10 + 119x/10 - 6**

**Problem 3 :**

-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)**

**= (x^{2}** + 5x)[x

**= (x^{2}** + 5x)(x

**= (x^{2}** + 5x)[(x

**= (x^{2}** + 5x)[(x

**= (x^{2}** + 5x)(x

**= x^{4}** + 4x

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