# Cubic Functions Derivatives

• May 14th 2008, 02:53 AM
andrew2322
Cubic Functions Derivatives
Hello all, i need assistance with these, thanks

1.) Find a cubic function whose graph touches the x-axis at x = -4, passes through the origin and has a value of 10 when x = 5.

2.) The function f(x) = 2x^3 + ax^2 - bx +3. When f(x) is divided by x - 2 the remainder is 15 and f(1) = 0

A.) Calculate the values of a and b
B.) Find the other two factors of f(x)

3.) The polynomial P(x) = x^3 + ax^2 + bx - 9 has zeros at x = 1 and x = -3. The values of a and b are?

this isnt homework, its revision so you dont have to worry about me cheating. THANKS!
• May 14th 2008, 04:36 AM
earboth
Quote:

Originally Posted by andrew2322
Hello all, i need assistance with these:

1.) Find a cubic function whose graph touches the x-axis at x = -4, passes through the origin and has a value of 10 when x = 5.
...

The general equation of a cubic function is:

$\displaystyle f(x)=ax^3+bx^2+cx+d$ and it's first derivative

$\displaystyle f'(x)=3ax^2+2bx+c$

$\displaystyle \begin{array}{l}f(0)=0 \\ f(-4)=0 \\ f'(-4)=0 \\ f(5)=10\end{array}$ ...... $\displaystyle ~\Longrightarrow~$...... $\displaystyle \begin{array}{l}d=0 \\ -64a+16b-4c =0 \\ 48a - 8b + c=0 \\ 125a+25b+5c=10\end{array}$

Solve the system of simultaneous equations.
• May 14th 2008, 04:51 AM
earboth
Quote:

Originally Posted by andrew2322
Hello all, i need assistance with these...

2.) The function f(x) = 2x^3 + ax^2 - bx +3. When f(x) is divided by x - 2 the remainder is 15 and f(1) = 0

A.) Calculate the values of a and b
B.) Find the other two factors of f(x)

...

to A.):

From f(1) = 0 you know that

$\displaystyle a-b+5=0$...... [1]

Use long divison to calculate the remainder. I've got as remainder r = 19 +4a - 2b which is according to your problem 15:

$\displaystyle 19+4a-2b = 15$ ...... [2]

Solve this system of simultaneous equations for a and b. I've got a = 3 and b = 8

Thus the equation of the function becomes:

$\displaystyle f(x)=2x^2+3x^2-8x+3$

to B.):

Use synthetic division :

$\displaystyle (2x^2+3x^2-8x+3) \div (x-1) = 2x^2+5x-3$

$\displaystyle 2x^2+5x-3 = 0$
You should get $\displaystyle x = \frac12~\vee~x=-3$ and therefore the other 2 factors are:
$\displaystyle (x+3)\left(x-\frac12\right)$