No worries, happy to help! To find P(3), we need to first determine the equation for the cubic polynomial P.
The general form of a cubic polynomial is:
P(x) = ax^3 + bx^2 + cx + d
Given that the leading coefficient (coefficient of the x^3 term) is 2, we have:
P(x) = 2x^3 + bx^2 + cx + d
We are also given that the coefficient of the linear term (coefficient of the x term) is -5, so we can rewrite the equation as:
P(x) = 2x^3 + bx^2 - 5x + d
Now, we can find the values of b, c, and d by using the additional information provided.
Since P(0) = 7, we substitute x = 0 into the equation:
P(0) = 2(0)^3 + b(0)^2 - 5(0) + d
Simplifying further, we have:
7 = d
Next, we use the fact that P(2) = 21 to solve for b and c.
P(2) = 2(2)^3 + b(2)^2 - 5(2) + d
Simplifying, we get:
21 = 16 + 4b - 10 + d
11 = 4b + d
To find P(3), we substitute x = 3 into the equation:
P(3) = 2(3)^3 + b(3)^2 - 5(3) + d
From the equations above, we know that d = 7 and 11 = 4b + d. Plugging these values into the equation, we have:
P(3) = 2(3)^3 + b(3)^2 - 5(3) + 7
P(3) = 54 + 9b - 15 + 7
P(3) = 56 + 9b
So, to find P(3), we need to determine the value of b. We can use the equation 11 = 4b + d to solve for b. Substituting d = 7, we have:
11 = 4b + 7
4b = 4
b = 1
Now we can substitute b = 1 into P(3) = 56 + 9b:
P(3) = 56 + 9(1)
P(3) = 56 + 9
P(3) = 65
Therefore, P(3) = 65.