The function given in the image is:

$f(x) = x^3 + 2x^2 - 4x - 8$

Let's work through the steps:

1. Finding the y-intercept

The y-intercept occurs when $x = 0$. So, we substitute $x = 0$ into the function:

$f(0) = (0)^3 + 2(0)^2 - 4(0) - 8 = -8$

The y-intercept is $-8$, which is already given in the problem.

2. Finding the rational roots (using the Rational Root Theorem)

The Rational Root Theorem helps us find potential rational roots of the polynomial. It states that any rational root is of the form $\frac{p}{q}$, where:

• $p$ is a factor of the constant term (-8 in this case).
• $q$ is a factor of the leading coefficient (1 in this case, since the coefficient of $x^3$ is 1).

The factors of -8 (the constant term) are: $\pm 1, \pm 2, \pm 4, \pm 8$

Since the leading coefficient is 1, the possible rational roots are: $\pm 1, \pm 2, \pm 4, \pm 8$

Now we test these values by substituting them into the function:

• $f(1) = (1)^3 + 2(1)^2 - 4(1) - 8 = 1 + 2 - 4 - 8 = -9$ (Not a root)
• $f(-1) = (-1)^3 + 2(-1)^2 - 4(-1) - 8 = -1 + 2 + 4 - 8 = -3$ (Not a root)
• $f(2) = (2)^3 + 2(2)^2 - 4(2) - 8 = 8 + 8 - 8 - 8 = 0$ (Root found: $x = 2$)
• $f(-2) = (-2)^3 + 2(-2)^2 - 4(-2) - 8 = -8 + 8 + 8 - 8 = 0$ (Root found: $x = -2$)

3. Factoring the polynomial

Since $x = 2$ and $x = -2$ are roots, we can factor the polynomial accordingly:

$f(x) = (x - 2)(x + 2)(x + 2)$

Or simplified as:

$f(x) = (x - 2)(x + 2)^2$

4. Identifying all real roots

From the factorized form $f(x) = (x - 2)(x + 2)^2$, we can see that the real roots are:

• $x = 2$ (with multiplicity 1)
• $x = -2$ (with multiplicity 2)

Thus, the real roots are: $x = 2, x = -2$

Would you like further details on sketching the graph or the sign table?

Here are 5 related questions to expand your understanding:

1. What is the significance of the multiplicity of a root in a polynomial?
2. How do we interpret the shape of the graph at roots with multiplicity greater than 1?
3. Can you determine if the function has any complex roots?
4. What role does the Rational Root Theorem play in finding irrational roots?
5. How can synthetic division be used to confirm the factorization of the polynomial?

Tip: When using the Rational Root Theorem, always test small values first, as they are easier to calculate and often yield results faster.