Example

3(r+2s)=2t-4
a⋅(b-2)=3b
3(r+2s)=2t-4
a⋅(b-2)=3b

Question

$$\frac{ { a }^{ 2 } - { b }^{ 2 } }{ a-b } - \frac{ { a }^{ 3 } - { b }^{ 3 } }{ { a }^{ 2 } - { b }^{ 2 } }$$

Answer

$$a+b-(a^2+a*b+b^2)/(a+b)$$

Solution


Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\frac{(a+b)(a-b)}{a-b}-\frac{{a}^{3}-{b}^{3}}{(a+b)(a-b)}\]
Use Difference of Cubes: \({a}^{3}-{b}^{3}=(a-b)({a}^{2}+ab+{b}^{2})\).
\[\frac{(a+b)(a-b)}{a-b}-\frac{(a-b)({a}^{2}+(a)(b)+{b}^{2})}{(a+b)(a-b)}\]
Remove parentheses.
\[\frac{(a+b)(a-b)}{a-b}-\frac{(a-b)({a}^{2}+ab+{b}^{2})}{(a+b)(a-b)}\]
Cancel \(a-b\).
\[a+b-\frac{(a-b)({a}^{2}+ab+{b}^{2})}{(a+b)(a-b)}\]
Cancel \(a-b\).
\[a+b-\frac{{a}^{2}+ab+{b}^{2}}{a+b}\]
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