Chapter 4.6, Problem 43WE

To determine

To express: the given as a polynomial

Answer to Problem 43WE

The polynomial equivalent to the given expression is $\boxed{x^4-6x^3+19x^2-30x+25}$

Explanation of Solution

Given information: The square of $x^2-3x+5$

Concept Involved:

Write the given description as a mathematical expression, then distribute the trinomial as first step, then distribute each term further, then group and combine like terms to get the final answer.

Calculation: Write the given description “The square of $x^2-3x+5$ ” as mathematical expression

  ${\left(x^2-3x+5\right)}^2=\left(x^2-3x+5\right)\left(x^2-3x+5\right)$

Distribute the binomial $\left(x^2-3x+5\right)$

  $x^2\left(x^2-3x+5\right)-3x\left(x^2-3x+5\right)+5\left(x^2-3x+5\right)$

Distribute further

  $x^4-3x^3+5x^2-3x^3+9x^2-15x+5x^2-15x+25$

Group like terms

  $x^4-3x^3-3x^3+5x^2+9x^2+5x^2-15x-15x+25$

Combine like terms

  $x^4-6x^3+19x^2-30x+25$

Conclusion:

Multiplying $\left(y-2x\right)\left(2x^2+y^2-3xy\right)$ we end up getting $\boxed{x^4-6x^3+19x^2-30x+25}$