What are the zeros of the function y = x^3 + 3x^2 – 4?

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The zeros of a function are the points where the graph intersects the x-axis, which means the output of the function (y) is zero. To find the zeros of the function y = x^3 + 3x^2 – 4, we need to set the equation equal to zero and solve for x:

x^3 + 3x^2 – 4 = 0

This equation is a polynomial of degree 3. To solve it, we can use various methods, such as factoring, the Rational Root Theorem, or numerical methods. Let’s try to find rational roots using the Rational Root Theorem:

The possible rational roots could be factors of the constant term (-4) divided by factors of the leading coefficient (1). This gives us the possible roots: ±1, ±2, and ±4.

Now we can test these possible roots:

  • x = 1:
    y = (1)^3 + 3(1)^2 – 4 = 1 + 3 – 4 = 0
    (1 is a root)
  • x = -1:
    y = (-1)^3 + 3(-1)^2 – 4 = -1 + 3 – 4 = -2
    (not a root)
  • x = 2:
    y = (2)^3 + 3(2)^2 – 4 = 8 + 12 – 4 = 16
    (not a root)
  • x = -2:
    y = (-2)^3 + 3(-2)^2 – 4 = -8 + 12 – 4 = 0
    (-2 is a root)
  • x = 4:
    y = (4)^3 + 3(4)^2 – 4 = 64 + 48 – 4 = 108
    (not a root)
  • x = -4:
    y = (-4)^3 + 3(-4)^2 – 4 = -64 + 48 – 4 = -20
    (not a root)

From our tests, we found that the roots are x = 1 and x = -2. To find the third root, we can use polynomial long division to divide the original polynomial by the known roots.

Dividing x^3 + 3x^2 – 4 by (x – 1), we get:

x^3 + 3x^2 - 4 = (x - 1)(x^2 + 4x + 4)

Next, we can factor (x^2 + 4x + 4):

x^2 + 4x + 4 = (x + 2)(x + 2) = (x + 2)^2

This gives us:

y = (x - 1)(x + 2)^2

Thus, the zeros of the function are:

  • x = 1 (crosses the x-axis)
  • x = -2 (touches the x-axis, as it is a double root)

In summary, the function y = x^3 + 3x^2 – 4 has zeros at x = 1 and x = -2.

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