Math Problem Statement

3, 1, -sqrt2

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

Solution

To create a polynomial function with the given zeros 3,1,23, 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 (xzero)(x - \text{zero}):

  • For 33, the factor is (x3)(x - 3).
  • For 11, the factor is (x1)(x - 1).
  • For 2-\sqrt{2}, the factor is (x+2)(x + \sqrt{2}).

Step 2: Account for the irrational root.

Because the polynomial must have rational coefficients, the conjugate of the irrational root 2-\sqrt{2} (which is 2\sqrt{2}) must also be a root. Thus, you also have the factor (x2)(x - \sqrt{2}).

Step 3: Write the polynomial as a product of all factors.

The polynomial is: f(x)=(x3)(x1)(x+2)(x2)f(x) = (x - 3)(x - 1)(x + \sqrt{2})(x - \sqrt{2})

Step 4: Simplify the product involving the irrational roots.

Notice that: (x+2)(x2)=x22(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)=(x3)(x1)(x22)f(x) = (x - 3)(x - 1)(x^2 - 2)

Step 6: Expand the polynomial.

First, multiply (x3)(x1)(x - 3)(x - 1): (x3)(x1)=x24x+3(x - 3)(x - 1) = x^2 - 4x + 3

Now multiply this by (x22)(x^2 - 2): f(x)=(x24x+3)(x22)f(x) = (x^2 - 4x + 3)(x^2 - 2)

Step 7: Perform the expansion.

f(x)=x2(x22)4x(x22)+3(x22)f(x) = x^2(x^2 - 2) - 4x(x^2 - 2) + 3(x^2 - 2) =x42x24x3+8x+3x26= x^4 - 2x^2 - 4x^3 + 8x + 3x^2 - 6

Step 8: Combine like terms.

f(x)=x44x3+(3x22x2)+8x6f(x) = x^4 - 4x^3 + (3x^2 - 2x^2) + 8x - 6 [

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Functions
Rational Coefficients
Conjugate Roots

Formulas

Difference of squares formula: (x + sqrt{a})(x - sqrt{a}) = x^2 - a
Polynomial expansion

Theorems

Conjugate Root Theorem
Fundamental Theorem of Algebra

Suitable Grade Level

Grades 10-12