Question

How do you find all the zeros of $ 2{x^3} + 9{x^2} + 6x - 8? $

Answer 1

Hint: As we know that in this question we have to use the rational roots theorem to find all the zeroes of the given polynomial function. We know that the rational roots theorem states that if the polynomial has integer coefficients, then the possible roots or zeroes are the factors of the constant term divided by the factors of the leading coefficient.
We can say any rational root of the polynomial are expressible in the form $ \dfrac{p}{q} $ for integers $ p,q $ where the integer $ p $ is a divisor of the constant term and $ q $ is the divisor of the coefficient term of the leading term.


Complete step by step solution:
According to the question we have the polynomial $ f(x) = 2{x^3} + 9{x^2} + 6x - 8 $ .
From the rational roots theorem we can say that the constant term here is $ p = - 8 $ . And $ q $ is the divisor of the coefficient $ 2 $ of the leading term.
This means we have the only possible rational roots: $ \pm \dfrac{1}{2}, \pm 1, \pm 2, \pm 4, \pm 8 $ .
Now we put $ x = - 2 $ and it gives us
 $ f( - 2) = 2(-8) + 9(4) + 6( - 2) - 8 = 0 $ .
So we can say that $ x = - 2 $ is a zero and $ (x + 2) $ is a factor. So we can take this common out of the polynomial and we can write
 $ (x + 2)(2{x^2} + 5x - 4) $ .
Now we apply the quadratic formula
 $ \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $ to solve the other quadratic equation.
So we have $ a = 2,b = 5,c = - 4 $ .
By putting the values we have
 $ x = \dfrac{{ - 5 \pm \sqrt {{5^2} - 4(2)( - 4)} }}{{2 \times 2}}\\
 \Rightarrow \dfrac{1}{4}\left( { - 5 \pm \sqrt {25 + 32} } \right) $ .
On further solving we have $ \dfrac{1}{4}( - 5 \pm \sqrt {57} ) $ .
Hence the required answer is $ \dfrac{1}{4}( - 5 \pm \sqrt {57} ) $ .
So, the correct answer is “-2 , $ \dfrac{1}{4}( - 5 \pm \sqrt {57} ) $ ,”.

Note: Before solving this kind of question we should have the full knowledge of rational roots theorem and how to apply those. We should note that it is a very important theorem, it tells us that with the given polynomial function with integer or whole number coefficients, a list of possible solutions can be found by listing the factors of the constant or the last term, over the factors of the coefficient of the leading term.