Find a monic quartic polynomial $f(x)$ with rational coefficients whose roots include $x=2-3\sqrt 2$ and $x=1-\sqrt 3$. give your answer in expanded form.

Monic means the coefficient on the highest degree term (leading coefficient) is one.quartic means a polynomial with a degree of 4.rational coefficients mean that the coefficients must be rational numbers (no imperfect square roots, for our case).======irrational root theoremif  is a root of a polynomial with rational coefficients, where  is irrational number, then   is also a root of that polynomial.if you do not use this theorem, you will not have rational coefficients. and are conjugates to each other. multiplying them together produces a difference of squares, which rationalizes them. this theorem is also related to how the quadratic equation has the  symbol.======difference of squares======if  has a root of  then we should also have a root of  . therefore, these are factors of the polynomial: let , if we multiply these out: so  is a factor of the polynomial function .======if has a root of  then we should also have a root of  . therefore, these are also factors of the polynomial: let , if we multiply these out: so  is a factor of the polynomial function  .======these two factors are factors of the polynomial functionmultiply them together to expand: so we have a  monic, quartic polynomial with rational coefficients  with the equationconfirm answer by plugging  into the function. you should get 0.