Question

X = 1/(8-√60) , what is the value of x³-5x²+8x-4

Answer 1

Answer:

1/(8-√60)3-5×(1/8-√60)2+8×1/8-√60-4

Answer 2

Answer Key

By repeated division, we see that $x^3-5x^2+8x-4=(x-2)^3+(x-2)^2$.

Note that $x-2=\dfrac{\sqrt{15} }{2}$.

Solution

Repeated division.

If we repeatedly divide the polynomial by $x-2$, we get $x^3-5x^2+8x-4=(x-2)^3+(x-2)^2$. Try using synthetic division to show this.

Rationalizing the denominator.

Try rationalizing the denominator. We get $x=2+\dfrac{\sqrt{15} }{2}$.

Substitution.

If we substitute $x-2=\dfrac{\sqrt{15} }{2}$ into the polynomial, we get $(\dfrac{\sqrt{15} }{2} )^3+(\dfrac{\sqrt{15} }{2} )^2$.

The value of the above is $\dfrac{15\sqrt{15} }{8} +\dfrac{15}{4} =\boxed{\dfrac{30+15\sqrt{15} }{8} }$.