Question

write a polynomial function of least degree with the given zero. -1+ √ 2, √ 3

Answer 1

Answer:

f(x) = $x^{4}$ + 2x³ - 4x² - 6x + 3

Step-by-step explanation:

Note that radical zeros occur in conjugate pairs, thus

- 1 + $\sqrt{2}$ is a zero then - 1 - $\sqrt{2}$ is also a zero

$\sqrt{3}$ is a zero then - $\sqrt{3}$ is also a zero

Thus the corresponding factors are

(x - (- 1 + $\sqrt{2}$) ), (x - (- 1 - $\sqrt{2}$) ), (x - $\sqrt{3}$), (x - (- $\sqrt{3}$)), that is

(x + 1 - $\sqrt{2}$), (x + 1 + $\sqrt{2}$), (x - $\sqrt{3}$), (x + $\sqrt{3}$)

The polynomial is then the product of the roots

f(x) = (x + 1 - $\sqrt{2}$)(x + 1 + $\sqrt{2}$)(x - $\sqrt{3}$)(x + $\sqrt{3}$)

     = ((x + 1)² - ($\sqrt{2}$)²)((x² - ($\sqrt{3}$)²)

     = (x² + 2x + 1 - 2)(x² - 3)

     = (x² + 2x - 1)(x² - 3) ← distribute

     = $x^{4}$ - 3x² + 2x³ - 6x - x² + 3

     = $x^{4}$ + 2x³ - 4x² - 6x + 3