a polynomial function has a root of -7 with multiplicity 2, a root of -1 with multiplicity 1, a root of 2…

a polynomial function has a root of -7 with multiplicity 2, a root of -1 with multiplicity 1, a root of 2 with multiplicity 4, and a root of 4 with multiplicity 1. if the function has a positive leading coefficient and is of even degree, which statement about the graph is true?\nthe graph of the function is positive on (2, 4).\nthe graph of the function is negative on (4, ∞).\nthe graph of the function is positive on (-∞, -7).\nthe graph of the function is negative on (-7, -1).

a polynomial function has a root of -7 with multiplicity 2, a root of -1 with multiplicity 1, a root of 2 with multiplicity 4, and a root of 4 with multiplicity 1. if the function has a positive leading coefficient and is of even degree, which statement about the graph is true?\nthe graph of the function is positive on (2, 4).\nthe graph of the function is negative on (4, ∞).\nthe graph of the function is positive on (-∞, -7).\nthe graph of the function is negative on (-7, -1).

Answer

Explanation:

Step1: Determine the polynomial form

The polynomial (P(x)=a(x + 7)^2(x + 1)(x - 2)^4(x - 4)) where (a>0).

Step2: Analyze intervals using test - points

  • For (x\in(-\infty,-7)), let (x=-8). Then (P(-8)=a(-8 + 7)^2(-8 + 1)(-8 - 2)^4(-8 - 4)>0).
  • For (x\in(-7,-1)), let (x=-2). Then (P(-2)=a(-2 + 7)^2(-2 + 1)(-2 - 2)^4(-2 - 4)>0).
  • For (x\in(-1,2)), let (x=0). Then (P(0)=a(0 + 7)^2(0 + 1)(0 - 2)^4(0 - 4)<0).
  • For (x\in(2,4)), let (x = 3). Then (P(3)=a(3 + 7)^2(3 + 1)(3 - 2)^4(3 - 4)<0).
  • For (x\in(4,\infty)), let (x=5). Then (P(5)=a(5 + 7)^2(5 + 1)(5 - 2)^4(5 - 4)>0).

Answer:

The graph of the function is positive on ((-\infty,-7))