Select all of the following that are potential roots of p(x) = x⁴ - 9x² - 4x + 12?
0, ±2, ±4, ±9, ±1/2, ±3, ±6, ±12

Solution:

For any polynomial f(x), k is the root only if f(k) = 0

To find which of the given values is a root, let us apply the remainder theorem. To find the remainder in each case. If the remainder is 0, then it is a root.

p(x) = x⁴ - 9x² - 4x + 12

We have to evaluate p(x) for these values and see if the answer is zero

p (0) = (0)⁴ - 9 (0)² - 4 (0) + 12 = 12 is not a root of this polynomial.

p (-2) = (-2)⁴ - 9 (-2)² - 4 (-2) + 12 = 16 - 36 + 8 + 12 = 0 is a root of this polynomial.

p (2) = (2)⁴ - 9 (2)² - 4 (2) + 12 = 16 - 36 - 8 + 12 = -16 is not a root of this polynomial.

p (-4) = (-4)⁴ - 9 (-4)² - 4 (-4) + 12 = 256 - 144 + 16 + 12 = 140

p (4) = (4)⁴ - 9 (4)² - 4 (4) + 12 = 256 - 144 - 16 + 12 = 108

p (-9) = (-9)⁴ - 9 (-9)² - 4 (-9) + 12 = 6561 - 729 + 36 + 12 = 5880

p (9) = (9)⁴ - 9 (4)² - 4 (9) + 12 = 6561 - 729 - 36 + 12 = 5808

p (-1/2) = (-1/2)⁴ - 9 (-1/2)² - 4 (-1/2) + 12 = 0.0625 - 2.25 + 2 + 12 = 11.8125

p (1/2) = (1/2)⁴ - 9 (1/2)² - 4 (1/2) + 12 = 0.0625 - 2.25 - 2 + 12 = 7.8125

p (-3) = (-3)⁴ - 9 (-3)² - 4 (-3) + 12 = 81 - 81 + 12 + 12 = 24

p (3) = (3)⁴ - 9 (3)² - 4 (3) + 12 = 81 - 81 - 12 + 12 = 0

p (-6) = (-6)⁴ - 9 (-6)² - 4 (-6) + 12 = 1296 - 324 + 24 + 12 = 1008

p (6) = (6)⁴ - 9 (6)² - 4 (6) + 12 = 1296 - 324 - 24 + 12 = 960

p (-12) = (-12)⁴ - 9 (-12)² - 4 (-12) + 12 = 20736 - 1296 + 48 + 12 = 19500

p (12) = (12)⁴ - 9 (12)² - 4 (12) + 12 = 20736 - 1296 - 48 + 12 = 19404

Therefore, the potential roots of p(x) = x⁴ - 9x² - 4x + 12 are - 2 and 3.

Select all of the following that are potential roots of p(x) = x⁴ - 9x² - 4x + 12?

Summary:

The potential roots of p(x) = x⁴ - 9x² - 4x + 12 is -2.