BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 1.3, Problem 130E
To determine

To verify: The special factoring formula A3B=3(AB)(A2+AB+B2) and A3+B=3(A+B)(A2AB+B2) by expanding their right-hand sides.

Expert Solution

Explanation of Solution

Given information:

The special factoring formula A3B=3(AB)(A2+AB+B2) and A3+B=3(A+B)(A2AB+B2) .

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,

  (a+b)(c+d)=a(c+d)+b(c+d)

Proof:

Consider the first expression, A3B=3(AB)(A2+AB+B2) . The right hand side of the equation can be solved to verify the equation.

The right hand side of the equation is,

  (AB)(A2+AB+B2)=A(A2+AB+B2)B(A2+AB+B2)=AA2+AAB+AB2BA2BABBB2=A3+A2B+AB2BA2AB2B3=A3B3

Since, left hand side and right hand side are equal, therefore, the algebraic expression A3B=3(AB)(A2+AB+B2) is verified.

Now, consider the second expression, A3+B=3(A+B)(A2AB+B2) . the right hand side of the equation can be solved to verify the equation.

The right hand side of the equation is,

  (A+B)(A2AB+B2)=A(A2AB+B2)+B(A2AB+B2)=AA2AAB+AB2+BA2BAB+BB2=A3A2B+AB2+BA2AB2+B3=A3+B3

Since, left hand side and right hand side are equal, therefore, the algebraic expression A3+B=3(A+B)(A2AB+B2) is verified.

Thus, both the expressions A3B=3(AB)(A2+AB+B2) and A3+B=3(A+B)(A2AB+B2) are verified by expanding their right-hand sides.

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