1. ## Polynomial Functions help please!!!

How would i write a polynomial equation from the roots 3 and 2+√7?

2. Originally Posted by peepsrock09
How would i write a polynomial equation from the roots 3 and 2+√7?
Note that $\displaystyle x=3$ is a zero, which implies $\displaystyle x-3$ is a factor of the polynomial

Note that $\displaystyle x=2+\sqrt{7}$ is a zero, which implies $\displaystyle x-\left(2+\sqrt{7}\right)$ is a factor of the polynomial. We also need the conjugate of our previous zero $\displaystyle x=2-\sqrt{7}$ to keep the coefficients clear of square roots. That implies that $\displaystyle x-\left(2-\sqrt{7}\right)$ is a factor of the polynomial.

Thus, your polynomial as a product of factors is $\displaystyle \left(x-3\right)\left(x-\left[2+\sqrt{7}\right]\right)\left(x-\left[2-\sqrt{7}\right]\right)$

Foil this out and see what you get.

Does this make sense?

3. I came up with x^3-8x^2+16x-3....

this is correct? or did i foil wrong?

4. Originally Posted by peepsrock09
I came up with x^3-8x^2+16x-3....

this is correct? or did i foil wrong?
Unfortunately, this is not correct.

How did you foil it out?? If you show what you did, I can easily show you where you made your mistake.

5. i took that equation, foiled to get:
(x-3)[x^2-x(2-√ 7)-x(2+√ 7)+(2+√ 7)(2-√ 7)]
(x-3)(x^2-2x+√ 7x-x-2x+1-√ 7x)
(x-3)(x^2-5x+1)
=
x^3-8x^2+16x-3

I think i foiled wrong in the first place..

6. Originally Posted by peepsrock09
i took that equation, foiled to get:
(x-3)[x^2-x(2-√ 7)-x(2+√ 7)+(2+√ 7)(2-√ 7)]
(x-3)(x^2-2x+√ 7x-x-2x+1-√ 7x)
(x-3)(x^2-5x+1)
=
x^3-8x^2+16x-3

I think i foiled wrong in the first place..

A simpler approach:

$\displaystyle \left( x - \left[ 2 + \sqrt{7} \right] \right)$ $\displaystyle \left(x - \left[ 2 - \sqrt{7} \right] \right) = \left( [x - 2] - \sqrt{7} \right)$ $\displaystyle \left( [x-2] + \sqrt{7} \right)$

which has the form of the factorisation of the difference of two squares:

$\displaystyle (A - B)(A + B) = A^2 - B^2$

(and yes I am attempting the record of how many times I can use the word of in one sentence).

Identifying $\displaystyle A = (x - 2)$ and $\displaystyle B = \sqrt{7}$ you therefore have:

$\displaystyle (x - 2)^2 - \left( \sqrt{7} \right)^2 = x^2 - 4x + 4 - 7 = x^2 - 4x - 3$.

7. aaahhhh. thank you soo much