Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1and the zeros - 3 and 5 + i

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  f(x) = x³ -7x² -4x +78

Step-by-step explanation:

The complex zeros come in conjugate pairs. Each zero (p) gives rise to a factor of the form (x -p). The polynomial is ...

  f(x) = (x -(-3))(x -(5+i))(x -(5-i)) = (x +3)((x -5)² -i²)

  = (x +3)(x² -10x +25 +1) = (x³ -10x² +26x) +(3x² -30x +78)

  f(x) = x³ -7x² -4x +78

a negative exponent is translated as follows:

so, we have

pentagon

step-by-step explanation:

the coordinates   a(5, -2), b(3,-1),c(-4,-4) d(-3,8) and e(-1,4) forms the vertices of the polygon when they are connected in order.

in order to classify that which polygon is it with the given coordinates, first count the number of vertices and then count the number of sides it can make with these vertices.

since, the given coordinates have five vertices, therefore, it will make five sides. moreover, pentagon is regular in nature therefore it will be convex.

now, a polygon with five sides is known as pentagon, therefore, the given polygon is the pentagon.