Find the roots of the polynomial equation.2x^3 + 2x^2 – 19x + 20 = 0

Answer 1

To find the roots of the polynomial equation, we can use factoring, the quadratic formula, or another appropriate method.

Let's try to factor the equation first:

2x^3 + 2x^2 – 19x + 20 = 0

We can see that 1 is not a factor, so let's try -1.

By substituting -1 into the equation, we get:

2(-1)^3 + 2(-1)^2 – 19(-1) + 20 = 0
-2 + 2 + 19 + 20 = 0
39 = 0

The equation does not hold true, so -1 is not a root. Hence, we cannot factor the equation easily.

Now, let's try using the quadratic formula:

Given a polynomial equation ax^3 + bx^2 + cx + d = 0, the quadratic formula states that the roots can be found using the formula:

x = (-b ± √(b^2 - 4ac))/(2a)

For the equation 2x^3 + 2x^2 – 19x + 20 = 0, we have a = 2, b = 2, and c = -19.

Using the quadratic formula:

x = (-2 ± √(2^2 - 4*2*(-19)))/(2*2)
x = (-2 ± √(4 + 152))/(4)
x = (-2 ± √(156))/(4)
x = (-2 ± 2√(39))/(4)
x = (-1 ± √(39))/(2)

Therefore, the roots of the polynomial equation 2x^3 + 2x^2 – 19x + 20 = 0 are:
x = (-1 + √(39))/2 and x = (-1 - √(39))/2.

Answer 2

I think this is incorrect bot... maybe I do not understand though.

Answer 3

Apologies for the mistake. Let's try to find the roots again using a different method.

We can use factoring by grouping to find the roots of the polynomial equation.

2x^3 + 2x^2 – 19x + 20 = 0

First, we group the terms:

(2x^3 + 2x^2) – (19x – 20) = 0

Next, we factor out the greatest common factor from each group:

2x^2(x + 1) – 19(x – 1) = 0

Now, we have two separate factors:

2x^2(x + 1) = 19(x – 1)

We can set each factor equal to zero and solve for x:

2x^2 = 0 or (x + 1) = 0 or 19(x – 1) = 0

From the first equation, we find x = 0.

From the second equation, we find (x + 1) = 0, which gives us x = -1.

From the third equation, we find x – 1 = 0, which gives us x = 1.

Therefore, the roots of the polynomial equation 2x^3 + 2x^2 – 19x + 20 = 0 are x = 0, x = -1, and x = 1.