Chapter 4.6, Problem 48WE

To determine

To express: the given as a polynomial

Answer to Problem 48WE

The polynomial which describes the given statement is $\boxed{-x^2+5xy-y^2}$

Explanation of Solution

Given information: Subtract the product of $2x-y$ and $x-2y$ from $x^2+y^2$

Concept Involved:

Rewrite the given statement as meaningful mathematical expression, then multiply first outer inner last terms to expand the product of binomial, then distribute the negative and drop the second parenthesis, then group and combine like terms to get the final answer.

Calculation:

Rewrite the given statement as meaningful mathematical expression

  $x^2+y^2-\left(2x-y\right)\left(x-2y\right)$

Multiply first outer inner last terms

  $x^2+y^2-\left(2x^2-4xy-xy+2y^2\right)$

Distribute the negative and drop the parenthesis

  $x^2+y^2-2x^2+4xy+xy-2y^2$

Group like terms

  $x^2-2x^2+y^2-2y^2+4xy+xy$

Combine like terms

  $-x^2-y^2+5xy$

Conclusion:

The polynomial that represents the description “Subtract the product of $2x-y$ and $x-2y$ from $x^2+y^2$ ” is $\boxed{-x^2+5xy-y^2}$