Find the quadratic polynomial with rational coefficients whose one zero is root 3 -2
Answers
Answer:
In mathematics, a polynomial transformation consists of computing the polynomial whose roots are a given function of the roots of a polynomial. Polynomial transformations such as Tschirnhaus transformations are often used to simplify the solution of algebraic equations.
Simple examples
Translating the roots
Let
P(x)=a_{0}x^{n}+a_{1}x^((n-1))+\cdots +a_((n))
be a polynomial, and
\alpha _{1},\ldots ,\alpha _{n}
be its complex roots (not necessarily distinct).
For any constant c, the polynomial whose roots are
\alpha _{1}+c,\ldots ,\alpha _{n}+c
is
Q(y)=P(y-c)=a_{0}(y-c)^{n}+a_{1}(y-c)^((n-1))+\cdots +a_((n)).
If the coefficients of P are integers and the constant c={\frac {p}{q)) is a rational number, the coefficients of Q may be not integers, but the polynomial cn Q has integer coefficients and has the same roots as Q.
A special case is when {\displaystyle c={\frac {a_{1)){na_{0))}.} The resulting polynomial Q does not have any term in yn − 1.
Reciprocals of the roots
Let
P(x)=a_{0}x^{n}+a_{1}x^((n-1))+\cdots +a_((n))
be a polynomial. The polynomial whose roots are the reciprocals of the roots of P as roots is its reciprocal polynomial
Q(y)=y^{n}P\left({\frac {1}{y))\right)=a_{n}y^{n}+a_((n-1))y^((n-1))+\cdots +a_((0)).
Scaling the roots
Let
P(x)=a_{0}x^{n}+a_{1}x^((n-1))+\cdots +a_((n))
be a polynomial, and c be a non-zero constant. A polynomial whose roots are the product by c of the roots of P is
Q(y)=c^{n}P\left({\frac {y}{c))\right)=a_{0}y^{n}+a_{1}cy^((n-1))+\cdots +a_((n))c^{n}.
The factor cn appears here because, if c and the coefficients of P are integers or belong to some integral domain, the same is true for the coefficients of Q.
In the special case where c=a_{0}, all coefficients of Q are multiple of c, and {\frac {Q}{c)) is a monic polynomial, whose coefficients belong to any integral domain containing c and the coefficients of P. This polynomial transformation is often used to reduce questions on algebraic numbers to questions on algebraic integers.
Combining this with a translation of the roots by {\displaystyle {\frac {a_{1)){na_{0)))), allows to reduce any question on the roots of a polynomial, such as root-finding, to a similar question on a simpler polynomial, which is monic and does not have a term of degree n − 1. For examples of this, see Cubic function § Reduction to a depressed cubic or Quartic function § Converting to a depressed quartic.
Step-by-step explanation: