To determine the potential roots of the polynomial function p(x) = x^4 + 22x^2 + 16x + 12, we can use the Rational Root Theorem, which suggests that any rational root, in the form of a fraction p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
In this case, the leading coefficient (of x^4) is 1 and the constant term is 12.
Step 1: Find the factors
- Factors of 12: ±1, ±2, ±3, ±4, ±6, ±12
Step 2: Test the potential roots
We’ll test the possible rational roots by substituting them into the polynomial:
- For x = 1:
p(1) = 14 + 22(1)2 + 16(1) + 12 = 1 + 22 + 16 + 12 = 51 (not a root) - For x = -1:
p(-1) = (-1)4 + 22(-1)2 + 16(-1) + 12 = 1 + 22 – 16 + 12 = 19 (not a root) - For x = 2:
p(2) = 24 + 22(2)2 + 16(2) + 12 = 16 + 88 + 32 + 12 = 148 (not a root) - For x = -2:
p(-2) = (-2)4 + 22(-2)2 + 16(-2) + 12 = 16 + 88 – 32 + 12 = 84 (not a root) - For x = 3:
p(3) = 34 + 22(3)2 + 16(3) + 12 = 81 + 198 + 48 + 12 = 339 (not a root) - For x = -3:
p(-3) = (-3)4 + 22(-3)2 + 16(-3) + 12 = 81 + 198 – 48 + 12 = 243 (not a root) - For x = 4:
p(4) = 44 + 22(4)2 + 16(4) + 12 = 256 + 352 + 64 + 12 = 684 (not a root) - For x = -4:
p(-4) = (-4)4 + 22(-4)2 + 16(-4) + 12 = 256 + 352 – 64 + 12 = 556 (not a root) - For x = 6:
p(6) = 64 + 22(6)2 + 16(6) + 12 = 1296 + 792 + 96 + 12 = 2196 (not a root) - For x = -6:
p(-6) = (-6)4 + 22(-6)2 + 16(-6) + 12 = 1296 + 792 – 96 + 12 = 2004 (not a root) - For x = 12:
p(12) = 124 + 22(12)2 + 16(12) + 12 = 20736 + 3168 + 192 + 12 = 23976 (not a root) - For x = -12:
p(-12) = (-12)4 + 22(-12)2 + 16(-12) + 12 = 20736 + 3168 – 192 + 12 = 23976 (not a root)
Conclusion: After testing all potential rational roots from the factors of 12, we find that none of them are roots of the polynomial function p(x) = x4 + 22x2 + 16x + 12. Therefore, if roots exist, they may be irrational or complex, requiring further methods such as synthetic division or numerical approximation for deeper analysis.