Chapter 1.3, Problem 57E

To determine

To calculate: The simplified value of the expression $\left(\left(x-1\right)+x^2\right)\left(\left(x-1\right)-x^2\right)$ .

Answer to Problem 57E

The value of the expression $\left(\left(x-1\right)+x^2\right)\left(\left(x-1\right)-x^2\right)$ is $-x^4+x^2-2x+1$ .

Explanation of Solution

Given information:

The expression $\left(\left(x-1\right)+x^2\right)\left(\left(x-1\right)-x^2\right)$ .

Formula used:

The special product formula of difference of squares to multiply the algebraic expressions is mathematically expressed as,

  $\left(A+B\right)\left(A-B\right)=A^2-B^2$

Special product of formula of perfect square of algebraic expressions which can be mathematically expressed as,

  ${\left(A-B\right)}^2=A^2-2AB+B^2$

Calculation:

Consider the given expression $\left(\left(x-1\right)+x^2\right)\left(\left(x-1\right)-x^2\right)$ .

Recall the special product formula of difference of squares to multiply the algebraic expressions is mathematically expressed as,

  $\left(A+B\right)\left(A-B\right)=A^2-B^2$

Here, $A=x-1$ and $B=x^2$ .

Apply it,

  $\left(\left(x-1\right)+x^2\right)\left(\left(x-1\right)-x^2\right)={\left(x-1\right)}^2-{\left(x^2\right)}^2$

Now, recall the special product of formula of perfect square of algebraic expressions which can be mathematically expressed as,

  ${\left(A-B\right)}^2=A^2-2AB+B^2$

  $\begin{array}{c}\left(\left(x-1\right)+x^2\right)\left(\left(x-1\right)-x^2\right)={\left(x-1\right)}^2-{\left(x^2\right)}^2\\ =x^2-2\left(x\right)\left(1\right)+1^2-x^4\\ =-x^4+x^2-2x+1\end{array}$

Thus, the value of the expression $\left(\left(x-1\right)+x^2\right)\left(\left(x-1\right)-x^2\right)$ is $-x^4+x^2-2x+1$ .