if a and b are the roots of x^2+x+1=0, then find the equation with roots a^19 and b^7. the ans is x^2+x+=0...now try to find a way to get this ans

Question

if a and b are the roots of  x^2+x+1=0, then find the equation with roots a^19 and b^7. the ans is x^2+x+=0...now try to find a way to get this ans

anonymous · Answer

$$a+b=-1$$
$$ab=1$$
solve a and b
$$\huge \color{purple}{ \text{ welcome ROSE MATTHEW}}$$

anonymous · Answer

this is because in a quadratic equation with roots$$\alpha,\beta \text{    and form   } ax^2+bx+c,\implies \alpha+\beta=\frac{ -b }{ a },\alpha \beta=\frac{ c }{ a }$$
then we see that
$$a(-1-a)=1$$
$$a^2+a+1=0$$

anonymous · Answer

Let's first factor the given equation:$$\bf x^2+x+1=0 \implies x=\frac{ -1 \pm \sqrt{1-4(1)(1)} }{ 2 }=\frac{ -1 \pm \sqrt{3}i}{ 2 }$$ So those are the two roots, now just make the polynomial as:$$\bf f(x)=(x-a^{19})(x-b^7)$$@rose.mathew

anonymous · Answer

@rose.mathew

anonymous · Answer

Show that $$ a^{19}=a\ 
b^7=b
$$
and you are with your original equation.

anonymous · Answer

Thinking with groups, the set {1,a,b} is a group. Notice that
$$
a=a\
a^2=b\
a^3=1\
b=b\
b^2=a\
b^3=1\
a b=1


$$