3, 1, -sqrt2


Write a polynomial function f of least degree, that has rational coefficients, a leading coefficient of 1, and the given zeros

To create a polynomial function with the given zeros \( 3, 1, -\sqrt{2} \), with rational coefficients and a leading coefficient of 1, follow these steps:

### Step 1: Express the zeros as factors.
For each zero, there is a corresponding factor of the form \( (x - \text{zero}) \):
- For \( 3 \), the factor is \( (x - 3) \).
- For \( 1 \), the factor is \( (x - 1) \).
- For \( -\sqrt{2} \), the factor is \( (x + \sqrt{2}) \).

### Step 2: Account for the irrational root.
Because the polynomial must have rational coefficients, the conjugate of the irrational root \( -\sqrt{2} \) (which is \( \sqrt{2} \)) must also be a root. Thus, you also have the factor \( (x - \sqrt{2}) \).

### Step 3: Write the polynomial as a product of all factors.
The polynomial is:
\[
f(x) = (x - 3)(x - 1)(x + \sqrt{2})(x - \sqrt{2})
\]

### Step 4: Simplify the product involving the irrational roots.
Notice that:
\[
(x + \sqrt{2})(x - \sqrt{2}) = x^2 - 2
\]
This is the difference of squares.

### Step 5: Multiply all the factors.
Now, the polynomial becomes:
\[
f(x) = (x - 3)(x - 1)(x^2 - 2)
\]

### Step 6: Expand the polynomial.
First, multiply \( (x - 3)(x - 1) \):
\[
(x - 3)(x - 1) = x^2 - 4x + 3
\]

Now multiply this by \( (x^2 - 2) \):
\[
f(x) = (x^2 - 4x + 3)(x^2 - 2)
\]

### Step 7: Perform the expansion.
\[
f(x) = x^2(x^2 - 2) - 4x(x^2 - 2) + 3(x^2 - 2)
\]
\[
= x^4 - 2x^2 - 4x^3 + 8x + 3x^2 - 6
\]

### Step 8: Combine like terms.
\[
f(x) = x^4 - 4x^3 + (3x^2 - 2x^2) + 8x - 6
\]
\[


Write a polynomial function f of least degree, that has rational coefficients, a leading coefficient of 1, and the given zeros: 3, 1, -sqrt2

Learn how to write a polynomial function with rational coefficients and zeros 3, 1, and -sqrt2. This step-by-step solution covers polynomial functions, conjugate root theorem, and expansion techniques. Suitable for 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