# Thread: polynomial division + simultaneous equations + other :S

1. ## polynomial division + simultaneous equations + other :S

Sorry about ridiculous title, but i wasnt sure ....

1) it is given that $\displaystyle f(x)= xcubed+3xsquared-6x-8$ (having large problems with the maths layout thing)

hence express $\displaystyle f(x)$ as a product of 3 linear factors

2)Solve $\displaystyle 4a+2b+12=0$and$\displaystyle a-b+3=0$

The 0's are really throwing me on this one :S

2. 1. $\displaystyle f(x) = x^3 + 3x^2 - 6x - 8$

Try factoring out $\displaystyle x - 2$.

$\displaystyle x^2(x - 2) = x^3 - 2x^2$

$\displaystyle x^3 + 3x^2 - 6x - 8$
$\displaystyle -(x^3 - 2x^2)$
$\displaystyle = 5x^2 - 6x - 8$

$\displaystyle 5x(x - 2) = 5x^2 - 10x$

$\displaystyle 5x^2 - 6x - 8$
$\displaystyle -(5x^2 - 10x)$
$\displaystyle = 4x - 8$

$\displaystyle 4(x - 2) = 4x - 8$

$\displaystyle 4x - 8$
$\displaystyle -(4x - 8)$
$\displaystyle = 0$

$\displaystyle \frac{x^3 + 3x^2 - 6x - 8}{x - 2} = x^2 + 5x + 4$

$\displaystyle x^3 + 3x^2 - 6x - 8 = (x - 2)(x^2 + 5x + 4) = (x - 2)(x + 1)(x + 4)$

2. Change the system to
$\displaystyle 4a + 2b = -12$
$\displaystyle a - b = -3$.

Multiplying the second equation by 2 yields
$\displaystyle 4a + 2b = -12$
$\displaystyle 2a - 2b = -6$

Adding those two equations yields
$\displaystyle 6a = -18$.

Can you finish?

3. 1st one = wow! so complicated

2nd one = i feel stupid :P

Thanks so much

4. The process I used for solving the first problem is known as long division of polynomials (and a little bit of factoring). For polynomials of degree 3 or greater, you will have to try dividing the polynomial by different factors using either this method or synthetic division, which is basically a symbolic representation of long division.