High School

Find the standard form of the function [tex]f(x) = -5x(x + 3)(x - 2)^2[/tex]. Then identify the y-intercept as an ordered pair.

A) Standard form: [tex]-5x^4 - 30x^3 + 60x^2[/tex]; Y-intercept: (0, 0)

B) Standard form: [tex]-5x^5 - 10x^4 + 10x^3[/tex]; Y-intercept: (0, -120)

C) Standard form: [tex]-5x^4 - 15x^3 + 15x^2[/tex]; Y-intercept: (0, 0)

D) Standard form: [tex]-5x^5 - 30x^4 + 60x^3[/tex]; Y-intercept: (0, -120)

Answer :

Final answer:

The standard form of the function f(x) = -5x(x + 3)(x - 2)^2 is -5x^3 +15x^2. The y-intercept of the function is at the ordered pair (0, 0).

Explanation:

To find the standard form of the function f(x) = -5x(x + 3)(x - 2)^2, we first need to expand the expression:

  1. Begin by expanding the term (x - 2)^2. This becomes x^2 - 4x + 4.
  2. Next, let's expand -5x(x + 3)(x^2 - 4x + 4) by distributing the terms. This gives us -5x^3 + 20x^2 - 20x - 5x^2 + 20x - 20.
  3. Finally, combine the like-terms to get: -5x^3 +15x^2 -60x +60.

So the standard form of the function is -5x^3 +15x^2. Next, to find the y-intercept, substitute x = 0 into the function. This gives f(0) = -5(0)^3 + 15(0)^2 = 0. So the y-intercept is the ordered pair (0, 0).

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