Chapter 1.3, Problem 130E

To determine

To verify: The special factoring formula $A^3-B=\left(A-B\right)\left(A^2+AB+B^2\right)$ and $A^3+B=\left(A+B\right)\left(A^2-AB+B^2\right)$ by expanding their right-hand sides.

Explanation of Solution

Given information:

The special factoring formula $A^3-B=\left(A-B\right)\left(A^2+AB+B^2\right)$ and $A^3+B=\left(A+B\right)\left(A^2-AB+B^2\right)$ .

Formula used:

To multiply two polynomial expressions, first multiply each term of first polynomial by second polynomial and then add the results. This property is known as distributive property, which is mathematically expressed as,

  $\left(a+b\right)\left(c+d\right)=a\left(c+d\right)+b\left(c+d\right)$

Proof:

Consider the first expression, $A^3-B=\left(A-B\right)\left(A^2+AB+B^2\right)$ . The right hand side of the equation can be solved to verify the equation.

The right hand side of the equation is,

  $\begin{array}{c}\left(A-B\right)\left(A^2+AB+B^2\right)=A\left(A^2+AB+B^2\right)-B\left(A^2+AB+B^2\right)\\ =A\cdot A^2+A\cdot AB+A\cdot B^2-B\cdot A^2-B\cdot AB-B\cdot B^2\\ =A^3+A^{2}B+A{B}^2-B{A}^2-A{B}^2-B^3\\ =A^3-B^3\end{array}$

Since, left hand side and right hand side are equal, therefore, the algebraic expression $A^3-B=\left(A-B\right)\left(A^2+AB+B^2\right)$ is verified.

Now, consider the second expression, $A^3+B=\left(A+B\right)\left(A^2-AB+B^2\right)$ . the right hand side of the equation can be solved to verify the equation.

The right hand side of the equation is,

  $\begin{array}{c}\left(A+B\right)\left(A^2-AB+B^2\right)=A\left(A^2-AB+B^2\right)+B\left(A^2-AB+B^2\right)\\ =A\cdot A^2-A\cdot AB+A\cdot B^2+B\cdot A^2-B\cdot AB+B\cdot B^2\\ =A^3-A^{2}B+A{B}^2+B{A}^2-A{B}^2+B^3\\ =A^3+B^3\end{array}$

Since, left hand side and right hand side are equal, therefore, the algebraic expression $A^3+B=\left(A+B\right)\left(A^2-AB+B^2\right)$ is verified.

Thus, both the expressions $A^3-B=\left(A-B\right)\left(A^2+AB+B^2\right)$ and $A^3+B=\left(A+B\right)\left(A^2-AB+B^2\right)$ are verified by expanding their right-hand sides.