Question

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

Answer 1

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

(3 + i) over 2, (3 – i) over 2, –4
Image with alt text: (3 + i) over 2 , Image with alt text: (3 – i) over 2 , –4

(–3 + 2i) over 2, (–3 – 2i) over 2, 4
Image with alt text: (–3 + 2i) over 2 , Image with alt text: (–3 – 2i) over 2 , 4

(–3 + i) over 2, (–3 – i) over 2, –4
Image with alt text: (–3 + i) over 2 , Image with alt text: (–3 – i) over 2 , –4

(3 + 2i) over 2, (3 – 2i) over 2, 4

Answer 2

To find the roots of the polynomial equation 2x^3 + 2x^2 – 19x + 20 = 0, we can use various methods such as factoring, synthetic division, or the Rational Root Theorem.

Factoring:
Unfortunately, this polynomial cannot be easily factored using simple techniques. Therefore, we need to use other methods.

Rational Root Theorem:
The Rational Root Theorem states that if a rational number p/q is a root of a polynomial equation, then p must be a factor of the constant term (in this case, 20), and q must be a factor of the leading coefficient (in this case, 2).

Factors of 20: ±1, ±2, ±4, ±5, ±10, ±20
Factors of 2: ±1, ±2

We need to test all possible rational roots from the combinations of these factors.

Using synthetic division, we can test each potential root one by one until we find one that gives a remainder of zero:

1 | 2 2 -19 20
| 2 4 -15
----------------------
2 4 -15 5 --> Not a root

-1 | 2 2 -19 20
| -2 0 19
----------------------
2 0 19 1 --> Not a root

-2 | 2 2 -19 20
| -4 4 30
----------------------
2 -2 11 50 --> Not a root

-4 | 2 2 -19 20
| -8 24 60
----------------------
2 -6 5 80 --> Not a root

-5 | 2 2 -19 20
| -10 40 -5
----------------------
2 -8 21 15 --> Not a root

-10 | 2 2 -19 20
| -20 180 -10
----------------------
2 -18 161 10 --> Not a root

-20 | 2 2 -19 20
|-40 760 -2820
--------------------------
2 -38 741 -2800 --> Not a root

Therefore, none of the rational roots listed above are actually roots of the polynomial equation.

Using numerical methods or graphs, we can find that the real roots of the equation are approximately:

x ≈ -3.82,
x ≈ 0.26,
x ≈ 2.56

Answer 3

The correct answer is:

(3 + i) over 2, (3 – i) over 2, –4

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