Question

Hi! Lets see if anyone can do this!

Answer 1

In order to factor a polynomial $P(x)$, you have to find its roots $x_1,x_2,\ldots,x_n$. Then, you can write

$P(x)=a(x-x_1)(x-x_2)\ldots(x-x_n)$

where $a$ is the leading coefficient of $P(x)$

a: Let's try to solve

$12u^2+5u+3=0$

the disciminant of this quadratic function is

$\Delta = b^2-4ac=25-144=-119$

So, this quadratic function has no roots, which implies that it cannot be factorised (using real numbers, at least. Are you allowing complex numbers? Let me know in that case)

b: Similarly, we set up

$50g^2 - 15g - 2=0$

This time we can find the two solutions

$g_1 = -\dfrac{1}{10},\quad g_2=\dfrac{2}{5}$

which yields the factorization

$50g^2 - 15g - 2=50\left(g+\dfrac{1}{10}\right)\left(g-\dfrac{2}{5}\right)$