Question

A polynomial function p (x) has degree 3, and its zeros are –3, 4, and 6. What do you think is the equation of p (x)? Do you think there could be more than one possibility? Explain

Answer 1

Answer:

$\bold{Cubic \: eq {}^{n} = x {}^{3} - ( \alpha + \beta + \gamma )x + ( \alpha \beta + \beta \gamma + \gamma \alpha )x - ( \alpha \beta \gamma )}$

Sum of zeroes ,

$\alpha + \beta + \gamma = ( - 3) + 4 + 6 = 7$

sum of product of zeroes taken two at a time ,

$\alpha \beta + \beta \gamma + \gamma \alpha = ( - 3)(4) + (4)(6) + (6)( - 3) \\ = > - 12 + 24 - 18 \\ = > - 6$

product of zeroes ,

$\alpha \beta \gamma = ( - 3)(4)(6) = - 72$

Substituting the values in the equation above , we get

$\bold{x {}^{3} - ( \alpha + \beta + \gamma )x {}^{2} + ( \alpha \beta + \beta \gamma + \gamma \alpha )x - ( \alpha \beta \gamma ) }\\\\\bold{ ⇢ x {}^{3} - 7x {}^{2} + ( - 6)x - ( - 72)} \\\\\bold\blue{ ⇢x {}^{3} - 7x {}^{2} - 6x + 72}$

hope helpful~