Question 1178029
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Real coefficients with roots 3 and 5i means -5i is another root.  Since the polynomial is degree 3, those are the only roots.<br>
The linear factor corresponding to the root 3 is (x-3); the quadratic factor corresponding to the roots 5i and -5i is (x^2+25).<br>
So f(x) is of the form<br>
{{{f(x) = a(x-3)(x^2+25)}}}<br>
Determine the constant a knowing that f(-1)=-312:<br>
{{{f(-1) = a(-4)(26) = -104a = -312}}}
{{{a -312/-104 = 3}}}<br>
The function is<br>
{{{f(x) = -3(x-3)(x^2+25)}}}
{{{f(x) = -3x^3+9x^2-75x+225}}}<br>