The number of roots common between the two equations is

Question

The number of roots common between the two equations

$x^3+3x^2+4x+7=0$ and $x^3+2x^2+7x+5=0$ is

$\color{green}{a.)\ 0 } \\~\\ b.)\ 1 \\~\\ c.)\ 2 \\~\\ d.)\ 3 \\~\\ $

i tried to solve both equations by subtracting then

$x^3+3x^2+4x+7-(x^3+2x^2+7x+5)=0 \\ x^2-3x+2=0 \\ x=2, \ 1$

but the answer is given as option $a.)$

I look for a short and simple way.

I have studied maths up to $12$th grade. Thanks!

Answer 1

You've found the $x$ values where the two expressions are equal. However, at neither of these $x$-values are the expressions equal to $0$, which is what you need for roots.

Answer 2

You can apply the Euclidean algorithm to the polynomials to find they're coprime, hence they can't have a common root (if they had a common root, say $\alpha$, $x-\alpha$ would be a common factor).

Answer 3

Considering $$f(x)=x^3+3x^2+4x+7$$ $$g(x)=x^3+2x^2+7x+5$$ you could notice that their derivatives never cancel in the real domain. So, $f(x)=0$ has only one real root and same for $g(x)=0$. So, the maximum number of common roots is $1$.

Now, inspection :

• $f(-3)=-5$, $f(-2)=3$; so the root for $f(x)=0$ is somewhere between $-3$ and $-2$.
• $g(-1)=-1$, $g(0)=5$; so the root for $g(x)=0$ is somewhere between $-1$ and $0$.

So, no common root.