If x - √5 is a factor of the cubic polynomial x^3 - 3√5x^2 + 13x - 3√5, then find all the zeroes of the polynomial.
Answers
Step-by-step explanation:
p(x) = x³ - 3√5x² + 13x - 3√5
x-√5 is a root of p(x)
We will divide p(x) by x-√5,
x-√5 ) x³ - 3√5x² + 13x - 3√5( x² - 2√5x + 3
x³ - √5x²
(-) (+)
-2√5x² + 13x
-2√5x² + 10x
(+) (-)
3x - 3√5
3x - 3√5
(-) (+)
0
Now we get a quadratic equation as quotient , we will find the roots of the quotient :
x² - 2√5x + 3
Roots = -b ± √(b²- 4ac)
2a
= -(-2√5) ± √[(-2√5)² - 4*1*3]
2*1
= 2√5 ± √(20 - 12)
2
= 2√5 ± √8
2
= 2√5 ± 2√2
2
√5 + √2 and √5 - √2
All the roots are √5 , √5 + √2 and √5 - √2.
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