Find a polynomial of degree 3 with real coefficients and zeros of minus​3,minus​1, and​ 4, for which ​f(minus​2)equals12.


To find a polynomial of degree 3 with real coefficients and given conditions, we'll follow these steps:

### Step 1: Start with the factored form
Since the polynomial has zeros at \(x = -3\), \(x = -1\), and \(x = 4\), the factored form of the polynomial can be written as:

\[
f(x) = a(x + 3)(x + 1)(x - 4)
\]

Here, \(a\) is a constant that we need to determine.

### Step 2: Use the condition \(f(-2) = 12\)
We're given that when \(x = -2\), the polynomial evaluates to 12, i.e., \(f(-2) = 12\). We can substitute \(x = -2\) into the polynomial and solve for \(a\).

\[
f(-2) = a((-2 + 3)(-2 + 1)(-2 - 4)) = 12
\]

Simplify the terms:

\[
f(-2) = a(1)(-1)(-6) = 12
\]

\[
f(-2) = 6a = 12
\]

### Step 3: Solve for \(a\)
Now, solve for \(a\):

\[
a = \frac{12}{6} = 2
\]

### Step 4: Write the polynomial
Now that we know \(a = 2\), the polynomial becomes:

\[
f(x) = 2(x + 3)(x + 1)(x - 4)
\]

### Step 5: Expand the polynomial
Let's expand the polynomial:

First, expand the first two factors:
\[
(x + 3)(x + 1) = x^2 + 4x + 3
\]

Now, expand this result with \((x - 4)\):
\[
(x^2 + 4x + 3)(x - 4) = x^3 - 4x^2 + 4x^2 - 16x + 3x - 12
\]
\[
= x^3 - 13x - 12
\]

Finally, multiply by 2:
\[
f(x) = 2(x^3 - 13x - 12) = 2x^3 - 26x - 24
\]

### Final answer:
The polynomial is:
\[
f(x) = 2x^3 - 26x - 24
\]

Would you like more details on the steps, or do you have any questions?

Here are 5 related questions to explore further:
1. How do you find the degree of a polynomial?
2. Can you solve for \(a\) in a different way?
3. What is the significance of the constant \(a\) in polynomial equations?
4. How would you factor the expanded polynomial back to its roots?
5. What happens if a polynomial has complex zeros?

**Tip:** When solving polynomial equations, always check the conditions given (like function values at certain points) to adjust the equation properly.

Find a polynomial of degree 3 with real coefficients and zeros of minus​3, minus​1, and​ 4, for which ​f(minus​2) equals 12.

Explore how to find a degree 3 polynomial with zeros at -3, -1, and 4, and satisfies f(-2) = 12. This problem covers polynomial factorization, solving for constants, and polynomial expansion. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation