Question
Factor by grouping. \[ x^{3}-2 x^{2}+7 x-14 \]
Ask by Higgins Rodriquez. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The expression can be factored as \((x-2)(x^{2}+7)\).
Solution
Factor the expression \( x^{3}-2x^{2}+7x-14 \).
Factor the expression by following steps:
- step0: Factor:
\(x^{3}-2x^{2}+7x-14\)
- step1: Rewrite the expression:
\(x^{3}+7x-2x^{2}-14\)
- step2: Factor the expression:
\(x\left(x^{2}+7\right)-2\left(x^{2}+7\right)\)
- step3: Factor the expression:
\(\left(x-2\right)\left(x^{2}+7\right)\)
The expression \(x^{3}-2x^{2}+7x-14\) can be factored by grouping as \((x-2)(x^{2}+7)\).
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Simplify this solution Bonus Knowledge
To factor the polynomial \( x^{3}-2 x^{2}+7 x-14 \) by grouping, we can group the terms as follows: \[ (x^3 - 2x^2) + (7x - 14) \] Now, factor out the common terms in each group: \[ x^2(x - 2) + 7(x - 2) \] We notice that both groups have a common factor of \( (x - 2) \): \[ (x - 2)(x^2 + 7) \] Thus, the fully factored form of \( x^{3}-2 x^{2}+7 x-14 \) is: \[ (x - 2)(x^2 + 7) \]
