Question

find a cubic polynomial with the sum sum of product of its zeroes taken two at a time and product of its zeros as 3, -1, -3 respectively​

Answer 1

Step-by-step explanation:

find cubic polynomial with the sum, sum of the product of its zeros taken two at a time, and product of its zeros as 5, -2, -24 respectively.

Found 2 solutions by josgarithmetic, ikleyn:

Answer by josgarithmetic(33072) (Show Source): You can put this solution on YOUR website!

Let these be the roots: p, r, v.

The description is this system:

system%28p%2Br%2Bv=5%2Cpr%2Brv%2Bpr=-2%2Cprv=-24%29

This is not a linear system, and solving it may take many steps. I have not gone further.

Answer by ikleyn(31566) (Show Source): You can put this solution on YOUR website!

.

find cubic polynomial with the sum, sum of the product of its zeros taken two at a time, and product of its zeros as 5, -2, -24 respectively.

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Let this polynomial be

p(x) = x%5E3+%2B+ax%5E2+%2B+bx+%2B+c = %28x-x%5B1%5D%29%2A%28x-x%5B2%5D%29%2A%28x-x%5B3%5D%29,

and x%5B1%5D, x%5B2%5D and x%5B3%5D are its roots.

Then, as you can check by performing multiplication and opening the parentheses,

a = -%28x%5B1%5D+%2B+x%5B2%5D+%2B+x%5B3%5D%29 = -5,

b = x%5B1%5D%2Ax%5B2%5D+%2B+x%5B1%5D%2Ax%5B3%5D+%2B+x%5B2%5D%2Ax%5B3%5D = -2,

c = -x%5B1%5D%2Ax%5B2%5D%2Ax%5B3%5D = 24.

So, the polynomial is p(x) = x%5E3+-+5x%5E2++-+2x+%2B+24.

Answer 2

$\huge\fbox \red{Solution:}$

Generally,

A cubic polynomial say, f(x) is of the form ax3 + bx2 + cx + d.

And, can be shown w.r.t its relationship between roots as.

⇒ f(x) = k [x3 – (sum of roots)x2 + (sum of products of roots taken two at a time)x – (product of roots)]

Where, k is any non-zero real number.

Here,

f(x) = k [x3 – (3)x2 + (-1)x – (-3)]

∴ f(x) = k [x3 – 3x2- x + 3)]

where, k is any non-zero real number.