Find the roots of the polynomial equation.
x^3 - 4x^2 + x + 26 = 0
A. 3 ± 2i, -2
B. -3 ± 2i, -2
C. 3 ± 2i, 2
D. -3 ± 2i, 2
Please help, I would appreciate it!
Thank you!
To find the roots of the polynomial equation x^3 - 4x^2 + x + 26 = 0, you can use various methods such as factoring, synthetic division, or the rational root theorem. However, the easiest method for this equation is to use the Rational Root Theorem.
The Rational Root Theorem states that if a polynomial equation has a rational root p/q, then p is a factor of the constant term (in this case, 26), and q is a factor of the leading coefficient (in this case, 1).
For the given equation, the constant term is 26 and the leading coefficient is 1. Therefore, the possible rational roots are the factors of 26 (±1, ±2, ±13, ±26).
By substituting each possible root into the equation, we can determine if it is indeed a root. By applying synthetic division with these possible roots, we can find that -2 is a root.
Using synthetic division, we can divide the polynomial equation by x + 2 (since -2 is a root) to find the remaining quadratic equation. The result of the synthetic division is given as follows:
-2 | 1 -4 1 26
-2 12 -26
_______________
1 -6 13 0
After performing the synthetic division, we obtained the quadratic equation x^2 - 6x + 13. To find the remaining roots, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For x^2 - 6x + 13, the coefficients are a = 1, b = -6, and c = 13. Substituting these values into the quadratic formula, we get:
x = (-(-6) ± √((-6)^2 - 4(1)(13))) / (2(1))
= (6 ± √(36 - 52)) / 2
= (6 ± √(-16)) / 2
Since the discriminant (b^2 - 4ac) is negative, the roots will be complex.
Simplifying further, we have:
x = (6 ± 4i) / 2
= 3 ± 2i
Therefore, the roots of the polynomial equation x^3 - 4x^2 + x + 26 = 0 are 3 ± 2i and -2. The correct answer is option A: 3 ± 2i, -2.
To find the roots of the polynomial equation x^3 - 4x^2 + x + 26 = 0, we can use a variety of methods such as factoring, synthetic division, or using a numerical method like Newton's method. In this case, let's use synthetic division to determine the possible rational roots of the equation.
To use synthetic division, we look for the factors of the constant term (26) and the leading coefficient (1). The factors of 26 are ±1, ±2, ±13, and ±26. The factors of 1 are ±1.
We can then set up a list of possible rational roots by dividing these factors.
±1/1, ±2/1, ±13/1, ±26/1
±1, ±2, ±13, ±26
Now, we perform synthetic division using each of these possible rational roots, one by one, until we find a root.
Using synthetic division, we find that when we divide by the possible root x = -2, the remainder is 0. This means that x = -2 is a root of the polynomial equation.
Performing synthetic division using the root x = -2:
```
-2 | 1 -4 1 26
| -2 12 -26
____________________
1 -6 13 0
```
The resulting polynomial after synthetic division is 1x^2 - 6x + 13. Now we have a quadratic equation to solve.
To find the remaining roots, we can use the quadratic formula or factoring. In this case, since the discriminant is negative, we will use the quadratic formula.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the quadratic equation 1x^2 - 6x + 13 = 0, a = 1, b = -6, and c = 13.
Using the quadratic formula, we can find the solutions:
x = (-(-6) ± √((-6)^2 - 4(1)(13))) / (2(1))
Simplifying:
x = (6 ± √(36 - 52)) / 2
x = (6 ± √(-16)) / 2
Since the square root of a negative number is not a real number, we can conclude that the remaining roots must be complex (involving the imaginary unit i).
Therefore, the roots of the polynomial equation x^3 - 4x^2 + x + 26 = 0 are -2, and 3 ± 2i.
Comparing the roots with the options given, we see that option B. -3 ± 2i, -2 matches the roots we found.
So the correct answer is B. -3 ± 2i, -2.
From the choices, clearly you expect 2 or -2 to be a root. A little synthetic division yields
(x+2)(x^2-6x+13)
So it's either A or B
Now use the quadratic formula to choose.