fill in the blanks so that the resulting statement is true.\nthe degree of the polynomial function (…

Explanation:

Step1: Find the degree of the polynomial

The degree of a polynomial (y = a(x - r_1)(x - r_2)\cdots(x - r_n)) is the sum of the exponents of the factors. For (f(x)=-2x^{3}(x - 1)(x + 5)), the exponent of the first factor (x^{3}) is (3), and for ((x - 1)) (which is (x^{1}-1)) the degree of the factor is (1), and for ((x + 5)) (which is (x^{1}+5)) the degree of the factor is (1). The degree of the polynomial (n=3 + 1+1).

Step2: Find the leading coefficient

First, expand ((x - 1)(x + 5)=x^{2}+5x-x - 5=x^{2}+4x - 5). Then (f(x)=-2x^{3}(x^{2}+4x - 5)). Using the distributive property (a(b + c+d)=ab+ac + ad), we have (f(x)=-2x^{3}\cdot x^{2}-2x^{3}\cdot4x+(-2x^{3})\cdot(- 5)). (f(x)=-2x^{5}-8x^{4}+10x^{3}).

Answer:

The degree of the polynomial function is (5). The leading coefficient is (-2).