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# ​ Please solve this, it seems I can't wrap my brain around it.

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1 My main problem is that I dont know write this polynomial's function.

Mar 8, 2021

### 1+0 Answers

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There are three turning points, so you have a fourth order polynomial.  It has roots at -5 and -1 (i..e. places where the function is zero.

Write the function as f(x) = -(x + 5)(x + 1)(x - a)(x - b)  where a and b are to be found (the initial negative sign is because the function goes to -infinity as x goes to +/- infinity)

When x = 0 we have f(0) = -25, so    -5*a*b = -25  ...(1)

There gradient of f is zero when x = -3, so differentiate f with respect to x and set the result equal to zero (with x = -3)

df/dx = -(4*x^3 + (18-3a-3b)*x^2 + (2ab-12a-12b+10)*x - 6ab - 5a -5b)

so 4*(-3)^3 + (18 - 3a - 3b)*(-3)^2 + (2ab -12a -12b + 10)*(-3) -6ab - 5a -5b = 0   ...(2)

You now have two equations for the two unknowns a and b.

Can you take it from here?

Mar 8, 2021
edited by Alan  Mar 8, 2021
edited by Alan  Mar 8, 2021