Given that root 2 is a zero of the cubic polynomial 6x3+2x2-10x-4root2, find its other two zeroes
LOL, should have checked that !
To find the other two zeroes of the cubic polynomial, we can use the fact that the sum of the zeroes of a cubic polynomial is equal to the negative coefficient of the quadratic term divided by the coefficient of the cubic term.
Given that √2 is a zero of the polynomial 6x³ + 2x² - 10x - 4√2, we can start by using synthetic division to divide the polynomial by (x - √2). This will leave us with a quadratic polynomial:
_____________________________
√2 | 6 2 -10 0
-12 6 -8
_____________________________
6 -10 -8 -8
The result of this synthetic division is the quadratic polynomial 6x² - 10x - 8. Now, we need to find the zeroes of this quadratic polynomial.
We can apply the quadratic formula to find the values of x. The quadratic formula is:
x = (-b ± √(b² - 4ac)) / (2a)
For the quadratic polynomial 6x² - 10x - 8, the coefficients are:
a = 6
b = -10
c = -8
Now, substituting these values into the quadratic formula:
x = (-(-10) ± √((-10)² - 4 * 6 * (-8))) / (2 * 6)
x = (10 ± √(100 + 192)) / 12
x = (10 ± √292) / 12
Therefore, the other two zeroes of the cubic polynomial 6x³ + 2x² - 10x - 4√2 are:
x₁ = (10 + √292) / 12
x₂ = (10 - √292) / 12
We know that (x-sqrt 2) is a factor
6 x^3 + 2 x^2 - 10 x -4 sqrt 2 = 0
(x-sqrt 2)( a x^2 + b x + c) = 0
well, c better be 4
(x-sqrt 2)(a x^2 + b x + 4) = 0
now I will use distributive property to split this up
x (a x^2 + b x + 4)
-sqrt 2 (a x^2 + b x + 4)
= 0
x terms:
4 x - b sqrt 2 x = 0
4 = b sqrt 2
b = 4/sqrt 2 = 2 sqrt 2
so now I have
x ( a x^2 + 2 sqrt 2 x + 4 )
-sqrt 2 (a x^2 + 2 sqrt 2 x + 4 ) = 0
so
x^2 terms
2 sqrt 2 x^2 - sqrt 2 a x^2 = 0
a = 2
so now I have
x ( 2 x^2 + 2 sqrt 2 x + 4 )
-sqrt 2 (2 x^2 + 2 sqrt 2 x + 4 ) = 0
in other words
(x-sqrt2)(2 x^2+2 sqrt 2 x + 4 = 0
so find zeros of
x^2 + sqrt 2 x + 2 = 0
x = [ -sqrt 2 +/- sqrt(2-32)] /2
= -(1/2) sqrt 2 +/- (i/2)sqrt 30
check my arithmetic, use for method only.
But √2 is not a root of
6x^3+2x^2-10x-4√2
6(√2)^3+2(√2)^2-10√2-4√2
= 6*2√2 + 2*2 - 10√2 - 4√2
= -2√2 + 4
Care to fix your mistake(s)?