If $y+\frac{1}{y}=5$, find in the simplest form the values of $y^3+\frac{1}{y^3}$

Question

If $y+\frac{1}{y}=5$, find in the simplest form the values of $y^3+\frac{1}{y^3}$

So I wrote:

$$y+\frac{1}{y}=5$$

with a common denominator which is:

$$\frac{y^2+1}{y}=5$$

Multiply $y$ to the other side and got:

$$y^2+1=5y$$

then i moved $5y$ and got

$$y^2+5y+1=0$$

but wouldnt that mean $y$ would have two values? Does anyone understand this problem?

Answer 1

Note that $(y+\frac{1}{y})^3=y^3+\frac{1}{y^3}+3y\frac{1}{y}(y+\frac{1}{y})$.

Answer 2

Squaring you get $$y^2+2+\frac{1}{y^2}=25$$

Thus $$y^2+\frac{1}{y^2}=23$$

Thus $$(y^2+\frac{1}{y^2})(y+\frac{1}{y})=23 \cdot 5$$

Do the multiplications.

Answer 3

Hint: Use the formula $$a^3+b^3=(a+b)(a^2+b^2-ab)$$. Find $a^2+b^2$ as mentioned in the comment