Question 1055571
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Find the nth-degree polynomial function with real coefficients satisfying the given conditions: 
n=4; -3; 1/4, 2i are {{{highlight(cross(zero))}}} zeroes and f(-2)=-144.
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A polynomial of the degree 4 with real coefficients and the roots -3, {{{1/4}}}, 2i is

f(x) = {{{a*(x-(-3))*(x-(1/4))*(x-2i)*(x+2i)}}} = {{{4a*(x+3)*(4x-1)*(x^2+4)}}}.     (1)


where "a" is the real number, a leading coefficient.

To find the value of "a", use the condition f(-2) = -144.

Substitute x= -2 into the polynomial (1). You will get

f(-2) = {{{4a*(-2+3)*(4*(-2)-1)*((-2)^2+4)}}} = {{{4a*1*(-9)*8}}} = -288a.

Thus  -288a = -144.  Hence,  a = {{{(-144)/(-288)}}} = 0.5.

Finally, the polynomial is

f(x) = {{{0.5*(x+3)*(4x-1)*(x^2+4)}}}.
</pre>