Answer:
The factors are (x - (1 - i√3)/2) , (x - (1 + i√3)/2) , (x - 3) , (x + 3) , (x - 1)
The zeroes are 1 , 3 , -3 , (1 - i√3)/2 , (1 + i√3)/2
Step-by-step explanation:
[tex](x^{5}+0-9x^{3} -x^{2}+0+9)[/tex] ÷ (x - 3) =
[tex](x^{4}-9x^{3}-x^{2}+0+9)[/tex] ÷ (x - 3) =
[tex]x^{4}+3x^{3}+(-x^{2}+0+9)[/tex] ÷ (x - 3) =
[tex]x^{4}+3x^{3}-x+(-3x+9)[/tex] ÷ (x - 3) =
[tex](x^{4}+3x^{3}-x-3)[/tex]
[tex](x^{4}+3x^{3}+0-x-3)[/tex] ÷ (x - 1) =
x³ + (4x³ + 0 - x - 3) ÷ (x - 1) =
x³ + 4x² + (4x² - x - 3) ÷ (x - 1) =
x³ + 4x² + 4x + (3x - 3) ÷ (x - 1) =
x³ + 4x² + 4x + 3
∵ -3 is a root ⇒ (x + 3) is a factor
(x³ + 4x² + 4x + 3) ÷ (x + 3) =
x² + (x² + 4x + 3) ÷ (x + 3) =
x² + x + (x + 3) ÷ (x + 3) =
x² + x + 1 ⇒ use the formula to find the factors of this quadratic
∵ a = 1 , b = 1 and c = 1
∴ [tex]x=\frac{-1+\sqrt{(1)^{2}-4(1)(1)} }{2(1)}=\frac{-1+\sqrt{-3}}{2}=\frac{-1+i\sqrt{3}}{2}[/tex]
∴ [tex]x=\frac{-1-i\sqrt{3} }{2}[/tex]
∴ The factors are (x - (1 - i√3)/2) , (x - (1 + i√3)/2) , (x - 3) , (x + 3) , (x - 1)
The zeroes:
x - 3 = 0 ⇒ x = 3
x + 3 = 0 ⇒ x = -3
x - 1 = 0 ⇒ x = 1
x - (1 - i√3)/2 = 0 ⇒ x = (1 - i√3)/2
x - (1 + i√3)/2 = 0 ⇒ x = (1 + i√3)/2
The zeroes are 1 , 3 , -3 , (1 - i√3)/2 , (1 + i√3)/2
