let a and b be real numbers where a ≠ b ≠ 0. which of the following functions could represent the graph…

let a and b be real numbers where a ≠ b ≠ 0. which of the following functions could represent the graph below? f(x)=x(x - a)^2(x - b)^4 f(x)=x(x - a)^3(x - b)^2 f(x)=x^4(x - a)(x - b)^2 f(x)=x^2(x - a)^5(x - b)

let a and b be real numbers where a ≠ b ≠ 0. which of the following functions could represent the graph below? f(x)=x(x - a)^2(x - b)^4 f(x)=x(x - a)^3(x - b)^2 f(x)=x^4(x - a)(x - b)^2 f(x)=x^2(x - a)^5(x - b)

Answer

Explanation:

Step1: Recall zero - multiplicity rules

If a factor of a polynomial is ((x - r)^n), when (n) is even, the graph touches the (x) - axis at (x = r), and when (n) is odd, the graph crosses the (x) - axis at (x = r).

Step2: Analyze the graph's behavior at (x = 0)

The graph crosses the (x) - axis at (x = 0), so the factor (x) (equivalent to ((x - 0))) must have an odd multiplicity.

Step3: Analyze the graph's behavior at other roots

The graph touches the (x) - axis at one of the non - zero roots and crosses at the other non - zero root. So one of ((x - a)) and ((x - b)) has an even multiplicity and the other has an odd multiplicity.

Step4: Check each option

  • For (f(x)=x(x - a)^2(x - b)^4), the multiplicities of (x), ((x - a)) and ((x - b)) are 1, 2 and 4 respectively. The graph touches the (x) - axis at (x=a) and (x = b) and crosses at (x = 0).
  • For (f(x)=x(x - a)^3(x - b)^2), the multiplicity of (x) is 1 (crosses at (x = 0)), multiplicity of ((x - a)) is 3 (crosses at (x=a)) and multiplicity of ((x - b)) is 2 (touches at (x = b)), which matches the graph's behavior.
  • For (f(x)=x^4(x - a)(x - b)^2), the multiplicity of (x) is 4 (touches at (x = 0)).
  • For (f(x)=x^2(x - a)^5(x - b)), the multiplicity of (x) is 2 (touches at (x = 0)).

Answer:

(f(x)=x(x - a)^3(x - b)^2)