Question content area top
Part 1
Form a polynomial whose real zeros and degree are given.
​Zeros: negative 2​,negative 1​,4​, 5​;    ​ degree: 4
Question content area bottom
Part 1
Type a polynomial with integer coefficients and a leading coefficient of 1.
​f(x)equals

enter your response here ​(Simplify your​ answer.)

To form a polynomial with given real zeros, you can create factors from each zero in the form \((x - \text{zero})\), then multiply these factors together. Given that the real zeros are \(-2\), \(-1\), \(4\), and \(5\), the corresponding factors are:

\[
(x + 2), (x + 1), (x - 4), (x - 5)
\]

Now, multiply these factors to form the polynomial:

\[
f(x) = (x + 2)(x + 1)(x - 4)(x - 5)
\]

### Step 1: Multiply the first two factors
\[
(x + 2)(x + 1) = x^2 + 3x + 2
\]

### Step 2: Multiply the last two factors
\[
(x - 4)(x - 5) = x^2 - 9x + 20
\]

### Step 3: Multiply the two results
\[
(x^2 + 3x + 2)(x^2 - 9x + 20)
\]

We will now expand this expression:

\[
x^2(x^2 - 9x + 20) + 3x(x^2 - 9x + 20) + 2(x^2 - 9x + 20)
\]

### Expanding each term:

1. \(x^2(x^2 - 9x + 20) = x^4 - 9x^3 + 20x^2\)
2. \(3x(x^2 - 9x + 20) = 3x^3 - 27x^2 + 60x\)
3. \(2(x^2 - 9x + 20) = 2x^2 - 18x + 40\)

### Combine like terms:
\[
f(x) = x^4 - 9x^3 + 20x^2 + 3x^3 - 27x^2 + 60x + 2x^2 - 18x + 40
\]

\[
f(x) = x^4 - 6x^3 - 5x^2 + 42x + 40
\]

Thus, the polynomial is:

\[
f(x) = x^4 - 6x^3 - 5x^2 + 42x + 40
\]

Would you like any more details or have any questions? 

Here are 5 related questions to consider:

1. How do you find the roots of a polynomial if only the equation is given?
2. How can you check if a given number is a root of a polynomial?
3. What is the relationship between the degree of a polynomial and its graph?
4. How can you factor a polynomial when the roots are complex?
5. How does the leading coefficient affect the behavior of a polynomial graph?

**Tip:** The degree of the polynomial determines the maximum number of real zeros or x-intercepts it can have.

Form a polynomial whose real zeros and degree are given. Zeros: negative 2, negative 1, 4, 5; degree: 4. Type a polynomial with integer coefficients and a leading coefficient of 1.

Learn how to form a degree 4 polynomial with integer coefficients and real zeros at -2, -1, 4, and 5. This solution covers key polynomial concepts such as factoring, multiplying, and expanding polynomials. Suitable for high school students (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