Hi

i want help in solving polynomial equation

2x^3-7x^2-15x=0

i know that the first part of the solution is

x(2 x^2+3x+-10 x-15)

but then i don't know how to complete solving the problem
i want detailed explanation to how to solve the problem

i also now the final answer is 0,-3/2 , 5

but i don't know how they found this solution

Question

Hi


i want help in solving polynomial equation

2x^3-7x^2-15x=0

i know that the first part of the solution is

x(2 x^2+3x+-10 x-15)

but then i don't know how to complete solving the problem
i want detailed explanation to how to solve the problem

i also now the final answer is 0,-3/2 , 5

but i don't know how they found this solution

kirbykirby · Answer

$$2x^3-7x^2-15x=0\
x(2x^2-7x-15)=0 $$
This means that either $x=0$ or $2x^2-7x-15 =0$
Notice that the part inside the parentheses is a polynomial of degree 2 which we know how to solve. 

You can try factoring it:
$ 2x^2-10x+3x-15=0\
2x(x-5)+3(x-5)=0\
(x-5)(2x+3)=0$
which means:
$x-5=0$ or $2x+3 = 0$

$\implies x=5, \text{ or }\
 2x=-3 \implies x=-3/2$

goformit100 · Answer

Hello, and A Warm Welcome to Open Study!

sam_pi · Answer

ok , in the beginning why did you place the -10 before the 3 ?

and here  2x(x-5)+3(x-5)=0 
how  did you factor this 

 did you divide the 10 /2 to get 5 and 15/3 to get 5 ?
why did you do that ?

kirbykirby · Answer

I broke up $-7x$ into the sum of $-10x $ and $3x$ 

Why do we do this? When you have a polynomial of the form $ax^2 + bx + c$, where $a$ is not just 1, then you find 2 numbers such that:
$m \times n = a \times c$ and
$m + n = b$

Your polynomial is $2x^2 - 7x - 15$, so 
$a=2 \b=-7 \c=-15$

So, you look for 2 numbers $m, n $ such that these will hold:
$m \times n = 2 \times (-15)=-30$
$ m + n = -7$
two such numbers that work are -10 and 3, since $-10 \times 3  = -30$ and $-10 + 3 = - 7$, so 
m = -10, n = 3

Now your polynomial $ax^2 + \color{red}{bx} + c$ becomes:
$ ax^2 + \color{red}{mx+ nx}+c $
$= 2x^2 \color{red}{-10 x + 3 x} -15 $

Now look at the first 2 terms, and the last two terms, and factor out what you can. THat's what I did in doing $2x(x-5)+3(x-5)$

Now to factor this out, let's look at a different but similar example. If you had:
$2x\color{blue}{y} + 3\color{blue}{y} $ I hope you can see that you can factor it as 
$\color{blue}{y}(2x+3$)

Now just go back to our original equation and substitute $y$ for $x-5$
$2x\color{blue}{(x-5)} + 3\color{blue}{(x-5)} $ factor it out the same way as we did above"
$\color{blue}{(x-5)}(2x+3$)