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    Find all zeros of the following polynomial. Write the polynomial in factored...

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labellegege041 | Student, Undergraduate | eNotes Newbie

Posted October 23, 2012 at 5:48 AM via web

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    Find all zeros of the following polynomial. Write the polynomial in factored form f(x)= x^3-27

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violy | High School Teacher | (Level 3) Assistant Educator

Posted January 3, 2013 at 5:04 AM (Answer #1)

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Let us equate the f(x) equal to zero,to find all the zeros of the given polynomial, hence, f(x) = 0:

x^3-27 = 0

We can use the formula below to factor the left-hand side.

 a^3 - b^3 = (a - b)(a^2 + ab + b^2)

So, x^3 - 3^3 = (x - 3)(x^2 + 3x +9)   <---this is the factored form of the polynomial

Then, we equate each factor to zero:

x - 3 = 0   and  x^2 + 3x + 9 = 0

For x - 3 = 0, we add 3 on both sides. 

 x - 3 + 3 =  0 + 3      ===>   x = 3   one of the zeros

We use Quadratic formula for the x^2 + 3x + 9 = 0.

a = 1, b = 3 and c = 9. 

So, x = [- 3 +/- sqrt((3)^2 - 4(1)(9))]/2(1)

===>  x = [-3 +/- sqrt(9 - 36)]/2

===> x = [-3 +/-sqrt(-27)]/2

===> x =[ -3 +/- sqrt(9)*sqrt(3)*sqrt(-1)]/2

===> x = [-3 +/- 3i sqrt(3)]/2

or other zeros are x = -3/2 + 3i sqrt(3)/2

 x = -3/2 - 3i sqrt(3)/2.

 

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