Question

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

Answer 1

To find the rational roots of the polynomial, we can use the rational root theorem. According to the theorem, any rational root of the polynomial must be of the form p/q, where p is a factor of the constant term -60 and q is a factor of 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.

Let's try all these possible rational roots and see which ones satisfy the equation:

p/q = ±1:
Checking p/q = 1: (1)^4 + 8(1)^3 + 7(1)^2 – 40(1) – 60 = 1 ≠ 0
Checking p/q = -1: (-1)^4 + 8(-1)^3 + 7(-1)^2 – 40(-1) – 60 = 1 ≠ 0

p/q = ±2:
Checking p/q = 2: (2)^4 + 8(2)^3 + 7(2)^2 – 40(2) – 60 = -32 ≠ 0
Checking p/q = -2: (-2)^4 + 8(-2)^3 + 7(-2)^2 – 40(-2) – 60 = -32 ≠ 0

p/q = ±3:
Checking p/q = 3: (3)^4 + 8(3)^3 + 7(3)^2 – 40(3) – 60 = -93 ≠ 0
Checking p/q = -3: (-3)^4 + 8(-3)^3 + 7(-3)^2 – 40(-3) – 60 = -93 ≠ 0

p/q = ±4:
Checking p/q = 4: (4)^4 + 8(4)^3 + 7(4)^2 – 40(4) – 60 = 524 ≠ 0
Checking p/q = -4: (-4)^4 + 8(-4)^3 + 7(-4)^2 – 40(-4) – 60 = 524 ≠ 0

p/q = ±5:
Checking p/q = 5: (5)^4 + 8(5)^3 + 7(5)^2 – 40(5) – 60 = 1885 ≠ 0
Checking p/q = -5: (-5)^4 + 8(-5)^3 + 7(-5)^2 – 40(-5) – 60 = 1885 ≠ 0

p/q = ±6:
Checking p/q = 6: (6)^4 + 8(6)^3 + 7(6)^2 – 40(6) – 60 = 2844 ≠ 0
Checking p/q = -6: (-6)^4 + 8(-6)^3 + 7(-6)^2 – 40(-6) – 60 = 2844 ≠ 0

p/q = ±10:
Checking p/q = 10: (10)^4 + 8(10)^3 + 7(10)^2 – 40(10) – 60 = 21440 ≠ 0
Checking p/q = -10: (-10)^4 + 8(-10)^3 + 7(-10)^2 – 40(-10) – 60 = 21440 ≠ 0

p/q = ±12:
Checking p/q = 12: (12)^4 + 8(12)^3 + 7(12)^2 – 40(12) – 60 = 59424 ≠ 0
Checking p/q = -12: (-12)^4 + 8(-12)^3 + 7(-12)^2 – 40(-12) – 60 = 59424 ≠ 0

p/q = ±15:
Checking p/q = 15: (15)^4 + 8(15)^3 + 7(15)^2 – 40(15) – 60 = 127665 ≠ 0
Checking p/q = -15: (-15)^4 + 8(-15)^3 + 7(-15)^2 – 40(-15) – 60 = 127665 ≠ 0

p/q = ±20:
Checking p/q = 20: (20)^4 + 8(20)^3 + 7(20)^2 – 40(20) – 60 = 228640 ≠ 0
Checking p/q = -20: (-20)^4 + 8(-20)^3 + 7(-20)^2 – 40(-20) – 60 = 228640 ≠ 0

p/q = ±30:
Checking p/q = 30: (30)^4 + 8(30)^3 + 7(30)^2 – 40(30) – 60 = 846540 ≠ 0
Checking p/q = -30: (-30)^4 + 8(-30)^3 + 7(-30)^2 – 40(-30) – 60 = 846540 ≠ 0

p/q = ±60:
Checking p/q = 60: (60)^4 + 8(60)^3 + 7(60)^2 – 40(60) – 60 = 7778880 ≠ 0
Checking p/q = -60: (-60)^4 + 8(-60)^3 + 7(-60)^2 – 40(-60) – 60 = 7778880 ≠ 0

Therefore, the rational roots of the polynomial x^4 + 8x^3 + 7x^2 – 40x – 60 = 0 are 2 and -6. The correct response is:

2, -6