Answer:
f(x) = [tex]x^{4}[/tex] + 2x³ - 4x² - 6x + 3
Step-by-step explanation:
Note that radical zeros occur in conjugate pairs, thus
- 1 + [tex]\sqrt{2}[/tex] is a zero then - 1 - [tex]\sqrt{2}[/tex] is also a zero
[tex]\sqrt{3}[/tex] is a zero then - [tex]\sqrt{3}[/tex] is also a zero
Thus the corresponding factors are
(x - (- 1 + [tex]\sqrt{2}[/tex]) ), (x - (- 1 - [tex]\sqrt{2}[/tex]) ), (x - [tex]\sqrt{3}[/tex]), (x - (- [tex]\sqrt{3}[/tex])), that is
(x + 1 - [tex]\sqrt{2}[/tex]), (x + 1 + [tex]\sqrt{2}[/tex]), (x - [tex]\sqrt{3}[/tex]), (x + [tex]\sqrt{3}[/tex])
The polynomial is then the product of the roots
f(x) = (x + 1 - [tex]\sqrt{2}[/tex])(x + 1 + [tex]\sqrt{2}[/tex])(x - [tex]\sqrt{3}[/tex])(x + [tex]\sqrt{3}[/tex])
= ((x + 1)² - ([tex]\sqrt{2}[/tex])²)((x² - ([tex]\sqrt{3}[/tex])²)
= (x² + 2x + 1 - 2)(x² - 3)
= (x² + 2x - 1)(x² - 3) ← distribute
= [tex]x^{4}[/tex] - 3x² + 2x³ - 6x - x² + 3
= [tex]x^{4}[/tex] + 2x³ - 4x² - 6x + 3
