Question

find the integral zeros of the polynomial 2 x^3 + 5 x^2 - 5 x - 2

Answer 1

Hey There!

Here we are given a polynomial. We have to find its zeros. So, we equate the polynomial to zero.

$2x^3+5x^2-5x-2=0$

Here, Sum of all coefficients of the polynomial is zero. This means that (x-1) is a factor.

$2x^3+5x^2-5x-2=0 \\ \\ \implies 2x^3-2x^2 + 7x^2-7x +2x -2=0 \\ \\ \implies 2x^2(x-1)+7x(x-1) + 2(x-1)=0 \\ \\ \implies (x-1)(2x^2+7x+2)=0 \\ \\ \implies \boxed{x=1} \, \, \,OR \, \, \, 2x^2+7x+2=0$


Now, $2x^2+7x+2=0$ is a quadratic equation.

Its Discriminant is:
$D=b^2-4ac =7^2 - 4(2)(2) = 33$
Since D is not a perfect square, the roots are irrational. Thus, two zeros of original polynomial are irrational.

There is only one integral zero, which is x=1


Hope it helps
Purva
Brainly Community

Answer 2

Answer:

2x³ + 5x² - 5x - 2

= 2x³ - 2x² + 7x² - 7x + 2x - 2

= 2x²(x - 1) + 7x(x - 1) + 2(x - 1)

= (2x² + 7x + 2)(x - 1)

Here we have to factors (2x² + 7x + 2) and (x - 1)

one zero of given Polynomial , x - 1 = 0 ⇒ x = 1 ( integer )

other two zeros of given polynomial , 2x² + 7x + 2 = 0 ⇒ x = (-7 ± √33)/4 [ irrational numbers ]

Hence, internal zero of polynomial has only one e.g., x = 1