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Difficulty: Very hardSimplifying and Factoring Algebraic Expressions

If aa and bb are distinct real numbers, what is the simplified form of the algebraic expression a3−b3−a(a2−b2)+b(a−b)2a−b\frac{a^3 - b^3 - a(a^2 - b^2) + b(a - b)^2}{a - b}?

  1. A
    ab+2b2ab + 2b^2
  2. B
    −ab-ab
  3. ababAnswer
  4. D
    a2+b2a^2 + b^2
  5. E
    a(a+b)a(a + b)

Answer

abab
Factoring the common binomial term (a−b)(a - b) from each term in the numerator yields (a−b)[(a2+ab+b2)−a(a+b)+b(a−b)](a - b)[(a^2 + ab + b^2) - a(a + b) + b(a - b)]. Expanding inside the brackets gives a2+ab+b2−a2−ab+ab−b2a^2 + ab + b^2 - a^2 - ab + ab - b^2, which reduces completely to abab. Dividing (a−b)(ab)(a - b)(ab) by (a−b)(a - b) leaves the simplified expression abab.

Step-by-Step Solution

1
Apply standard algebraic factoring identities to each component of the numerator.
a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2), −a(a2−b2)=−a(a−b)(a+b)-a(a^2 - b^2) = -a(a - b)(a + b), and b(a−b)2=b(a−b)(a−b)b(a - b)^2 = b(a - b)(a - b).
Rewriting each grouping with the common binomial factor (a−b)(a - b) allows for structured factoring of the numerator.
2
Factor out (a−b)(a - b) from the entire numerator.
Numerator =(a−b)[(a2+ab+b2)−a(a+b)+b(a−b)]= (a - b) \left[ (a^2 + ab + b^2) - a(a + b) + b(a - b) \right].
Extracting the common factor isolates the remaining polynomial terms inside brackets.
3
Expand and collect like terms within the bracketed expression.
(a2+ab+b2)−(a2+ab)+(ab−b2)=a2−a2+ab−ab+ab+b2−b2=ab(a^2 + ab + b^2) - (a^2 + ab) + (ab - b^2) = a^2 - a^2 + ab - ab + ab + b^2 - b^2 = ab.
Canceling opposite terms (a2−a2=0a^2 - a^2 = 0, ab−ab=0ab - ab = 0, b2−b2=0b^2 - b^2 = 0) leaves only abab inside the brackets.
4
Divide the factored numerator by the denominator (a−b)(a - b).
(a−b)(ab)a−b=ab\frac{(a - b)(ab)}{a - b} = ab.
Since aa and bb are distinct (a≠ba \neq b), a−b≠0a - b \neq 0, allowing non-zero cancellation.

Key Concept

Simplifying complex algebraic expressions by factoring common binomial terms and applying algebraic identities
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Topics

Simplifying and Factoring Algebraic Expressions

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