Let's break down and solve the problem shown in the image.

Problem:

Find a polynomial of degree 3 with real coefficients and zeros of $-3$, $-1$, and $4$, for which $f(-2) = 24$.

Step-by-Step Solution:

1. Form the Polynomial: Since the zeros of the polynomial are $-3$, $-1$, and $4$, the factors of the polynomial are: $(x + 3), (x + 1), (x - 4)$ The general form of the polynomial is: $f(x) = a(x + 3)(x + 1)(x - 4)$ where $a$ is a constant to be determined.

2. Use the Condition $f(-2) = 24$: To find $a$, we substitute $x = -2$ into the polynomial and set it equal to 24: $f(-2) = a(-2 + 3)(-2 + 1)(-2 - 4) = 24$ Simplifying each factor: $f(-2) = a(1)(-1)(-6) = 24$ $-6a = 24$ Solving for $a$: $a = \frac{24}{-6} = -4$

3. Write the Final Polynomial: Now that we know $a = -4$, the polynomial is: $f(x) = -4(x + 3)(x + 1)(x - 4)$

4. Expand the Polynomial (Optional): If you want to simplify the polynomial further, you can expand it: $f(x) = -4[(x + 3)(x + 1)(x - 4)]$ First, expand $(x + 3)(x + 1)$: $(x + 3)(x + 1) = x^2 + 4x + 3$ Now multiply this by $(x - 4)$: $(x^2 + 4x + 3)(x - 4) = x^3 - 4x^2 + 4x^2 - 16x + 3x - 12$ Simplify: $= x^3 - 13x - 12$ Finally, multiply by $-4$: $f(x) = -4(x^3 - 13x - 12) = -4x^3 + 52x + 48$ So the polynomial is: $f(x) = -4x^3 + 52x + 48$

This is the polynomial you're looking for. Do you need further clarification on any step?

Related Questions:

1. How do you find the roots of a cubic polynomial?
2. What is the importance of the constant $a$ in defining a polynomial?
3. Can the polynomial have complex roots if the coefficients are real?
4. What methods can be used to factor a cubic polynomial directly?
5. How does the transformation $f(-2) = 24$ affect the graph of the polynomial?

Tip:

When given roots, always form the polynomial using the corresponding factors and then adjust for any scaling conditions using additional information provided (like $f(x)$ values).