Question

show that 5+2,
$show \: that \: 5 + 2 \sqrt{7 \: s \: rrational}$
​

Answer 1

Take : $x=5+2\sqrt{7}$

To Prove : Use Rational Zero Theorem

Rational Zero Theorem

c-e here stands for co-efficient.

Rational solution of a polynomial is between

(±c-e of the Highest Degree Term) OVER

(±c-e of the Lowest Degree Term)

To use rational zero theorem,

we eliminate the roots by isolating & squaring.

$x-5 =2\sqrt{7}$

Square both sides.

$x^2-10x+25=28$

$x^2-10x-3=0$

According to the rational zero theorem

rational solution exists only between $\frac{\pm3}{\pm1}$.

In other words 3, -3, 1, -1.

But any of these are the zeros.

Therefore, the zero $5+2\sqrt{7}$ is not a rational.

And therefore, it is an irrational.

Hence shown.