SOLUTION: Write an equation for a cubic polynomial p(x) with leading coefficient -1 whose graph passes through the point (2, 8) and is tangent to the x-axis at the origin.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Write an equation for a cubic polynomial p(x) with leading coefficient -1 whose graph passes through the point (2, 8) and is tangent to the x-axis at the origin.       Log On


   

Question 707573: Write an equation for a cubic polynomial p(x) with leading coefficient -1 whose graph passes through the point (2, 8) and is tangent to the x-axis at the origin.
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
p%28x%29=-x%5E3%2Bbx%5E2%2Bcx%2Bd

derivative of p(0) must be zero because of tangent at the origin. Taking derivative, setting it equal zero, seems c=0.

p(2)=8, so carrying this through did not seem to help. 8=-8+4b+d, not too useful.

Point (0,0) must satisfy our p(x) polynomial, so 0=-0%5E3%2Bb%2A0%5E2%2Bd,
apparantly d=0.

So far we seem to have p%28x%29=-x%5E3%2Bbx%5E2.
Let's try point (2,8) again.
p%28x%29=-%28x%5E2%29%28x-b%29, also.
Using the point, 8=-%282%29%5E2%2A%282-b%29
.
b=4

THIS should work well: p%28x%29=-%28x%5E2%29%28x-4%29