Chapter 4.6, Problem 49WE

To determine

To find: the polynomial based on the given description

Answer to Problem 49WE

The polynomial which describes the given statement is $\boxed{x^3-2x^2-5x+6}$

Explanation of Solution

Given information: Polynomial is divided by $x-3$ , the quotient is $x^2+x-2$ .

Concept Involved:

Assign variable for the unknown that we need to find, based on the given statement replace the keywords to get a meaningful mathematical equation. Solve the equation.

Calculation: Let the polynomial be P

Based on the statement “Polynomial is divided by $x-3$ , the quotient is $x^2+x-2$ ” write an equation

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

To get P alone in one side of the equation Multiply $\left(x-3\right)$ on both sides of the equation

  $\left(x-3\right)\cdot \frac{P}{\left(x-3\right)}=\left(x-3\right)\left(x^2+x-2\right)$

Simplify the fraction in left side of the equation

  $\cancel{\left(x-3\right)}\cdot \frac{P}{\cancel{\left(x-3\right)}}=\left(x-3\right)\left(x^2+x-2\right)$

Distribute the binomial $\left(x-3\right)$ in right side of the equation

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

Distribute further to rewrite the right side of the equation

  $P=x^3-3x^2+x^2-3x-2x+6$

Combine like terms in right side of the equation

  $P=x^3-2x^2-5x+6$

Conclusion:

The polynomial that represents the description “Polynomial is divided by $x-3$ , the quotient is $x^2-x-2$ ” is $\boxed{x^3-2x^2-5x+6}$