Chapter 5.4, Problem 6MCQ

To determine

To Evaluate: The simplified form of the expression $\left(x+y\right)\left(x^2+2xy-y^2\right)$ where $x\ne 0,y\ne 0$ .

Answer to Problem 6MCQ

The value of the expression $\left(x+y\right)\left(x^2+2xy-y^2\right)$ is $x^3+3x^{2}y+x{y}^2-y^3$.

Explanation of Solution

Given information:

The expression is $\left(x+y\right)\left(x^2+2xy-y^2\right)$ .

Formula used:

Distribution property states that multiplication of a expression with sum or difference of two other terms same as add or subtract them separately after multiplication. That is

  $a\left(b\pm c\right)=ab\pm ac$

Rule for simplify algebraic expression is remove parentheses and then group all like terms.

Calculation:

Consider the expression is $\left(x+y\right)\left(x^2+2xy-y^2\right)$ .

Recall that distribution property states that multiplication of a expression with sum or difference of two other terms same as add or subtract them separately after multiplication. That is

  $a\left(b\pm c\right)=ab\pm ac$

Apply it,

  $\begin{array}{c}\left(x+y\right)\left(x^2+2xy-y^2\right)=\left(x+y\right)x^2+\left(x+y\right)\left(2xy\right)-\left(x+y\right)y^2\\ =\left(x\times x^2+y\times x^2\right)+\left(x\times 2xy+y\times 2xy\right)-\left(x\times y^2+y\times y^2\right)\\ =\left(x^3+y{x}^2\right)+\left(2x^{2}y+2x{y}^2\right)-\left(x{y}^2+y^3\right)\end{array}$

Again recall that rule for simplify algebraic expression is remove parentheses and then group all like terms.

Use it,

  $\begin{array}{c}\left(x^3+y{x}^2\right)+\left(2x^{2}y+2x{y}^2\right)-\left(x{y}^2+y^3\right)=x^3+y{x}^2+2x^{2}y+2x{y}^2-x{y}^2-y^3\\ =x^3+\left(y{x}^2+2x^{2}y\right)+\left(2x{y}^2-x{y}^2\right)-y^3\\ =x^3+3x^{2}y+x{y}^2-y^3\end{array}$

Thus, the value of the expression $\left(x+y\right)\left(x^2+2xy-y^2\right)$ is $x^3+3x^{2}y+x{y}^2-y^3$ .