Question

solve this ..................

Answer 1

1.
Proof:
Let $f(x)$ be a given polynomial function.
Then, by Polynomial remainder theorem, $(x-2)$ is a divisor (factor) of $f(x)$ if and only if $f(2)=0$.

Hence,
$f(2)=2\cdot 8 -9\cdot 4+7\cdot 2+6=16-36+14+6=0$

Q.E.D.

2. 
Since $(x-2)$ is a divisor of $f(x)=2x^3-9x^2+7x+6$, there is exists $g(x)$ such that $f(x)=(x-2)\cdot g(x)$.
Hence, 
$g(x)=f(x)/(x-2)$.
Therefore,
$g(x)=2x^2-5x-3$

Now, let's find the roots of g(x):
$\displaystyle x_{1,2}= \frac{5\pm \sqrt{25+24} }{4}= \frac{5\pm7}{4}= 3,- \frac{1}{2}$

Hence,

$x_1,x_2,x_3=3,- 0.5,2$ are solutions of f(x).