Respuesta :
correct answer is option d.
Step-by-step explanation:
Here, we have The following expression to check Which of the following is a polynomial with roots: −square root of 3 , square root of 3, and 2 :
a. x3 + 3x2 − 5x − 15
For [tex]-\sqrt{3}[/tex] , [tex]x^{3} + 3x^{2} -5x -15[/tex] ⇒ [tex](-\sqrt{3}) ^{3} + 3(-\sqrt{3} )^{2} -5(-\sqrt{3} ) -15[/tex] [tex]\neq 0[/tex] . So, this isn't the one. We move on further!
b. x3 + 2x2 − 3x − 6
For [tex]-\sqrt{3}[/tex] , [tex]x^{3} + 2x^{2} -3x -6[/tex] ⇒ [tex](-\sqrt{3}) ^{3} + 2(-\sqrt{3} )^{2} -3(-\sqrt{3} ) -6[/tex] = 0 .
For [tex]\sqrt{3}[/tex] , [tex]x^{3} + 2x^{2} -3x -6[/tex] ⇒ [tex](\sqrt{3}) ^{3} + 2(\sqrt{3} )^{2} -3(\sqrt{3} ) -6[/tex] = 0.
For 2 , [tex]x^{3} + 2x^{2} -3x -6[/tex] ⇒ [tex](2) ^{3} + 2(2)^{2} -3(2 ) -6[/tex][tex]\neq 0[/tex] . So, this isn't the one. We move on further!
c. x3 − 3x2 − 5x + 15
For [tex]-\sqrt{3}[/tex] , [tex]x^{3}- 3x^{2} -5x -15[/tex] ⇒ [tex](-\sqrt{3}) ^{3}- 3(-\sqrt{3} )^{2} -5(-\sqrt{3} ) -15[/tex] [tex]\neq 0[/tex] . So, this isn't the one. We move on further!
d. x3 − 2x2 − 3x + 6
For [tex]-\sqrt{3}[/tex] , [tex]x^{3} - 2x^{2} -3x +6[/tex] ⇒ [tex](-\sqrt{3}) ^{3} - 2(-\sqrt{3} )^{2} -3(-\sqrt{3} ) +6[/tex] = 0 .
For [tex]\sqrt{3}[/tex] , [tex]x^{3} - 2x^{2} -3x +6[/tex] ⇒ [tex](\sqrt{3}) ^{3} -2(\sqrt{3} )^{2} -3(\sqrt{3} ) +6[/tex] = 0.
For 2 , [tex]x^{3} - 2x^{2} -3x +6[/tex] ⇒ [tex](2) ^{3} - 2(2)^{2} -3(2 ) +6[/tex] = 0.
Therefore, correct answer is option d.
