Answer: x = -1, -3, 7 + √62, 7 - √62
Step-by-step explanation:
q p
x⁴ - 10x³ - 66x² - 94x - 39
[tex]\frac{p}{q}[/tex] = +/- [tex]\frac{1*3*13*39}{1}[/tex]
possible rational factors: 1, -1, 3, -3, 13, -13, 39, -39
Use synthetic division or long division to see which factor will leave a remainder of 0.
try x + 1 = 0 ⇒ x = -1
-1 | 1 -10 -66 -94 -39
| ↓ -1 11 55 39
1 -11 -55 -39 0
(x + 1)(x³ - 11x² - 55x - 39)
next, try x + 3 = 0 ⇒ x = -3 for the new polynomial
-3 | 1 -11 -55 -39
| ↓ -3 42 39
1 -14 -13 0
(x + 1)(x + 3)(x² - 14x - 13)
Lastly: find the zeros by setting each factor equal to zero and solve.
x + 1 = 0 ⇒ x = -1
x + 3 = 0 ⇒ x = -3
x² - 14x - 13 = 0
[tex]x = \frac{-(b) +/- \sqrt{(b)^{2} -4(a)(c)}}{2(a)}[/tex]
[tex]= \frac{-(-14) +/- \sqrt{(-14)^{2} -4(1)(-13)}}{2(1)}[/tex]
[tex]= \frac{14 +/- \sqrt{196+52}}{2}[/tex]
[tex]= \frac{14 +/- \sqrt{248}}{2}[/tex]
[tex]= \frac{14 +/- 2\sqrt{62}}{2}[/tex]
[tex]= 7 +/- \sqrt{62}[/tex]
