A quartic polynomial has 2 distinct real roots at x=1 and x=−3/5. If the function has a y-intercept at−1and has f(2)=2 and f(3)=3. How to determine the remaining roots when two distinct real roots and y-intercept are given.

Khalfanebw 2022-09-19 Answered
A quartic polynomial has 2 distinct real roots at x = 1 and x = 3 / 5. If the function has a y-intercept at 1 and has f ( 2 ) = 2 and f ( 3 ) = 3. How to determine the remaining roots when two distinct real roots and y-intercept are given.
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Answers (1)

Mackenzie Lutz
Answered 2022-09-20 Author has 2 answers
You can sum up everything you know in a system like this:
{ f ( 1 ) = 0 f ( 3 5 ) = 0 f ( 0 ) = 1 f ( 2 ) = 2 f ( 3 ) = 3
Then it's very simple to find all the coefficients of f ( x ) = a x 4 + b x 3 + c x 2 + d x + e, because you have a 5-variable system and 5 equations, so the solution is unique! For example, from the 3rd equation we get f ( 0 ) = 0 + 0 + 0 + 0 + e = 1 e = 1, and so on. After some calculus we obtain:
{ a = 5 156 b = 41 78 c = 289 156 d = 14 39 e = 1
Finally we manage to find the polynomail, and the associated function:
f ( x ) = 5 156 x 4 41 78 x 3 + 289 156 x 2 14 39 x 1
So all the real roots are x 1 = 3 5 , x 2 = 1 , x 3 = 8 2 3 , x 4 = 8 + 2 3 .
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