Answer:
To determine the missing coefficient "a" in the polynomial P(x) = x^4 - 3x^3 + ax^2 - 6x + 14, we can use the SOLVE process as follows:
S: Write down the given information and identify the problem.
We are given the polynomial P(x) = x^4 - 3x^3 + ax^2 - 6x + 14, and we need to find the value of the missing coefficient "a".
O: Organize the information and decide on a plan.
To find the value of "a", we can substitute specific values for x into the polynomial and solve for a.
L: Carry out the plan.
For example, let's say we substitute x = 2 into the polynomial. We get:
P(2) = 2^4 - 32^3 + a2^2 - 6*2 + 14
= 16 - 24 + 4a - 12 + 14
= -4 + 4a + 2
= 4a - 2
We are given that P(2) = 4a - 2 = 0, so 4a = 2.
Solving for a, we get:
a = 2 / 4
= 0.5
V: Check the solution.
We can check the solution by substituting 0.5 for a in the original polynomial and verifying that it gives us the correct result for P(x) when x = 2.
P(x) = x^4 - 3x^3 + 0.5x^2 - 6x + 14
Substituting x = 2 and a = 0.5, we get:
P(2) = 2^4 - 32^3 + 0.52^2 - 6*2 + 14
= 16 - 24 + 1 - 12 + 14
= 0
Since P(2) = 0, our solution appears to be correct.
Therefore, the missing coefficient "a" in the polynomial P(x) = x^4 - 3x^3 + ax^2 - 6x + 14 is 0.5.