Find a cubic polynomial whose zeroes are 3, 5 and -2.

1.	Find a cubic polynomial whose zeroes are 3, 5 and -2.
Answer» Let α, β and γ are the zeroes of the required polynomial. Then we have: α + β + γ = 3 + 5 + (-2) = 6 αβ + βγ + γα = 3 × 5 + 5 × (-2) + (-2) × 3 = -1 and αβγ = 3 × 5 × -2 = -30 Now, p(x) = x³ – x² (α + β + γ) + x (αβ + βγ + γα) – αβγ = x³ – x² × 6 + x × (-1) – (-30) = x³ – 6x² – x + 30 So, the required polynomial is p(x) = x³ – 6x² – x + 30.

Answer»

Let α, β and γ are the zeroes of the required polynomial.

Then we have:

α + β + γ = 3 + 5 + (-2) = 6

αβ + βγ + γα = 3 × 5 + 5 × (-2) + (-2) × 3 = -1

and αβγ = 3 × 5 × -2 = -30

Now, p(x) = x³ – x² (α + β + γ) + x (αβ + βγ + γα) – αβγ

= x³ – x² × 6 + x × (-1) – (-30)

= x³ – 6x² – x + 30

So,

the required polynomial is p(x) = x³ – 6x² – x + 30.

Discussion