if x minus 5 root 2 is a factor of cubic polynomial x cube minus 3 root 5 x square + 13 x minus 3 root 5 then find all the zeros of polynomial
Answers
given that x-√5 is a factor of the cubic polynomial x3-3√5x2+13x-3√5
x-√5 ) x3-3√5x2+13x-3√5 ( x2 -2√5x + 3
x3- √5x2 ( substract )
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- 2√5x2+13x
- 2√5x2+10x ( substract )
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3x - 3√5
3x - 3√5 ( substract )
------------------------
0
∴ The quotient is x2 -2√5x + 3 = 0
Using roots of quadratic formula
a = 1, b = 2√5, c = 3
x = (-b ± √(b2 - 4ac) ) / 2a
x = (2√5 ± √((2√5)2 - 12) ) / 2
∴ the other zeros are x = √5 ± √2.
Hope it helps you!
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Step-by-step explanation:
Hi friend,
Here is your answer,
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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.