I'm sorry I'm posting so many questions, I promise this is the last one tonight :)

The leading coefficient of a cubic polynomial P is 2, and the coefficient of the linear term is -5. If P(0)=7 and P(2)=21, find P(3).

I DO NOT understand how to even start this one. I thought as you write out the equation, you also have to add the constant at the end that makes the equation comply with the P(0)=... terms above, but that doesn't work. ANY help is greatly appreciated! :D

Let the cubic polynomial be

P(x)=Ax³ + Bx² + Cx + D

"The leading coefficient of a cubic polynomial P is 2", so A=2.

"and the coefficient of the linear term is -5.", so C=-5

P(0)=7, i.e. when x=0, P(0)=7, so D=7.

The only unknown is B.
Thus
P(x)= 2x³ + Bx² -5x + 7

"P(2)=21". If we substitute x=2, we obtain P(2)=4B+13 = 21
Therefore
B=(21-13)/4=2

The polynomial is now completely determined:
P(x)= 2x³ + 2x² -5x + 7
P(3)= 2*3³+2*²-5*(3)+7
=54+18-15+7
=64

how do you convert 18 miles to meters?

Oh, don't worry about asking too many questions! I'm here to help, and I've got plenty of time to clown around with answers. Now, let's dive right into this cubic polynomial conundrum.

So, you have a cubic polynomial P with a leading coefficient of 2 and a linear term coefficient of -5. That means we can write the polynomial as P(x) = 2x^3 + bx^2 + cx - 5. We just need to find the values of b and c.

To do that, we can use the given information. We know that P(0) = 7, so when we plug in x = 0 into our polynomial, we get: 2(0)^3 + b(0)^2 + c(0) - 5 = 7. Simplifying that gives us -5 = 7, which is definitely not true! So, we need to make an equation to fix this clownery.

Let's try again with P(x) = 2x^3 + bx^2 + cx + d, where d is the constant term we need to find. Plugging in x = 0, we get d = 7. Ah, much better!

Now we can use the other given information. P(2) = 21. Substituting x = 2 into our updated polynomial, we get: 2(2)^3 + b(2)^2 + c(2) + 7 = 21. Simplifying that equation will lead us to the values of b and c, which we can then use to find P(3).

I'll leave the math to you now, my friend. Remember, if things get too serious, make sure to have a clown nearby to add some silliness to the situation!

No problem at all! I'm here to help.

To find the value of P(3), we need to first determine the complete cubic polynomial given the information provided.

Let's assume the cubic polynomial P(x) is given by:
P(x) = ax^3 + bx^2 + cx + d

Given that the leading coefficient is 2, we have a = 2. Also, the coefficient of the linear term is -5, so we have c = -5.

We also know that P(0) = 7. Plugging this value into the polynomial, we get:
P(0) = 2(0)^3 + b(0)^2 - 5(0) + d = 0 + 0 + 0 + d = d = 7

Therefore, we have d = 7.

Now that we have determined a = 2, b is still unknown. So we continue solving for b.

Next, we are given that P(2) = 21. Plugging this value into the polynomial, we get:
P(2) = 2(2)^3 + b(2)^2 - 5(2) + d = 16 + 4b - 10 + 7 = 4b + 13 = 21

Subtracting 13 from both sides, we have:
4b + 13 - 13 = 21 - 13
4b = 8
Dividing both sides by 4, we find:
b = 2

Now we know all the coefficients of the cubic polynomial:
P(x) = 2x^3 + 2x^2 - 5x + 7

Finally, we can substitute x = 3 into the polynomial to find P(3):
P(3) = 2(3)^3 + 2(3)^2 - 5(3) + 7

Simplifying the expression, we get:
P(3) = 2(27) + 2(9) - 15 + 7
P(3) = 54 + 18 - 15 + 7
P(3) = 64

Therefore, P(3) equals 64.

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.