Final answer:
To make P(x) divisible by x, the constant term in P(x) must be zero. After simplifying P(x), we find that the constant term is 9 + 2c, which is set to zero. Solving for c, we get c = -4.5.
Step-by-step explanation:
To find the value of c such that P(x) is divisible by x, we should first simplify P(x) and then set the terms that would give the constant term when x equals zero. Since P(x) is divisible by x, the constant term has to be zero for P(x) to have x as a factor.
We have P(x) = (x - 3)^2 – 2(x – c). Expanding the squared term gives us x^2 - 6x + 9. Now, rewriting P(x), we get: P(x) = x^2 - 6x + 9 - 2x + 2c. Combining like terms, the equation now reads: P(x) = x^2 - (6 + 2)x + (9 + 2c). For the equation to be divisible by x, the constant term, which is 9 + 2c, must be zero. Thus, 9 + 2c = 0. Solving for c, we subtract 9 from both sides and get 2c = -9. Dividing both sides by 2, we find that c = -4.5.
Therefore, the value of c that makes P(x) divisible by x is -4.5.