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Chapter 6: Problem 73

Factor by grouping. $$x^{3}-2 x^{2}+5 x-10$$

Short Answer

Expert verified
The factored form of the expression is \((x^{2} + 5)(x - 2)\).

Step by step solution

01

Group Terms

Rearrange the expression in a way that makes the factorisation clearer. It may be beneficial to think in terms of pairs and arrange the expression: \(x^{3} - 2x^{2} + 5x - 10\). This can be re-written as two pairs: \(x^{3} - 2x^{2} + 5x - 10\).
02

Factor each group

For each of the pairs of terms, look for the greatest common factor. For the first pair, \(x^{3} - 2x^{2}\), the greatest common factor is \(x^{2}\). Factoring \(x^{2}\) out gives \(x^{2}(x - 2)\). Similarly for the second pair, \(5x - 10\), the common factor is 5, giving: \(5(x - 2)\). This gives the new expression: \(x^{2}(x - 2) + 5(x - 2)\).
03

Factor the entire expression

Now, the expression \(x^{2}(x - 2) + 5(x - 2)\) has a common factor across its two parts: \(x - 2\). Factoring out this term gives the final factorized expression: \((x^{2} + 5)(x - 2)\).

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