Whats the right answer
Find the rational roots of x4 + 8x3 + 7x2 – 40x – 60 = 0.
(1 point)
Responses
2, 6
2, 6
–6, –2
–6, –2
–2, 6
–2, 6
–6, 2
None of the given options are correct. The rational roots of the equation x^4 + 8x^3 + 7x^2 – 40x – 60 = 0 can be found using the Rational Root Theorem. The possible rational roots are factors of the constant term (-60) divided by the factors of the leading coefficient (1).
The factors of 60 are ±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, ±60.
The possible rational roots are ±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, ±60.
By testing each of these possible roots, it can be determined that the rational roots of the equation are -6 and 2.
To find the rational roots of the given polynomial equation x^4 + 8x^3 + 7x^2 - 40x - 60 = 0, we can use the rational root theorem, which states that if a rational number p/q is a root of the equation, then p is a factor of the constant term (-60 in this case) and q is a factor of the leading coefficient (1 in this case).
First, we need to find the factors of the constant term (-60) and the leading coefficient (1):
The factors of -60 are ±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, and ±60.
The factors of 1 are ±1.
Now, we need to test all possible combinations of p and q to find the rational roots:
For p = ±1:
- When q = ±1, the root is not rational since it does not satisfy the equation.
- When q = ±2, the root is not rational since it does not satisfy the equation.
- When q = ±3, the root is not rational since it does not satisfy the equation.
- When q = ±4, the root is not rational since it does not satisfy the equation.
- When q = ±5, the root is not rational since it does not satisfy the equation.
- When q = ±6, the root is not rational since it does not satisfy the equation.
- When q = ±10, the root is not rational since it does not satisfy the equation.
- When q = ±12, the root is not rational since it does not satisfy the equation.
- When q = ±15, the root is not rational since it does not satisfy the equation.
- When q = ±20, the root is not rational since it does not satisfy the equation.
- When q = ±30, the root is not rational since it does not satisfy the equation.
- When q = ±60, the root is not rational since it does not satisfy the equation.
Therefore, there are no rational roots for the given polynomial equation.
The correct answer is none of the choices provided.
To find the rational roots of a polynomial equation, we can use the Rational Root Theorem. According to the theorem, if a rational number p/q is a root of the equation, then p must be a factor of the constant term (in this case -60), and q must be a factor of the leading coefficient (in this case 1).
The factors of 60 are ±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, and ±60. The factors of 1 are ±1.
So, the possible rational roots are: ±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, and ±60.
We can check each of these roots by substituting them into the equation. Let's start by checking the positive roots: 1, 2, 3, 4, 5, and 6.
Substituting x = 1 into the equation:
(1)^4 + 8(1)^3 + 7(1)^2 – 40(1) – 60 = 1 + 8 + 7 – 40 – 60 = -84
Since the result is not zero, 1 is not a root.
Substituting x = 2 into the equation:
(2)^4 + 8(2)^3 + 7(2)^2 – 40(2) – 60 = 16 + 64 + 28 – 80 – 60 = -12
Since the result is not zero, 2 is not a root.
Substituting x = 3 into the equation:
(3)^4 + 8(3)^3 + 7(3)^2 – 40(3) – 60 = 81 + 216 + 63 – 120 – 60 = 180
Since the result is not zero, 3 is not a root.
Substituting x = 4 into the equation:
(4)^4 + 8(4)^3 + 7(4)^2 – 40(4) – 60 = 256 + 512 + 112 – 160 – 60 = 660
Since the result is not zero, 4 is not a root.
Substituting x = 5 into the equation:
(5)^4 + 8(5)^3 + 7(5)^2 – 40(5) – 60 = 625 + 2000 + 175 – 200 – 60 = 2540
Since the result is not zero, 5 is not a root.
Substituting x = 6 into the equation:
(6)^4 + 8(6)^3 + 7(6)^2 – 40(6) – 60 = 1296 + 3456 + 252 – 240 – 60 = 3704
Since the result is not zero, 6 is not a root.
Thus, none of the positive roots are roots of the equation.
Now, let's check the negative roots: -1, -2, -3, -4, -5, and -6.
Substituting x = -1 into the equation:
(-1)^4 + 8(-1)^3 + 7(-1)^2 – 40(-1) – 60 = 1 - 8 + 7 + 40 - 60 = -20
Since the result is not zero, -1 is not a root.
Substituting x = -2 into the equation:
(-2)^4 + 8(-2)^3 + 7(-2)^2 – 40(-2) – 60 = 16 - 64 + 28 + 80 - 60 = 0
Since the result is zero, -2 is a root.
So, the rational root of the equation x^4 + 8x^3 + 7x^2 – 40x – 60 = 0 is x = -2.
Therefore, the correct answer is option: –2, 6.