Question
Factor polynomial \( b^{3}-2 b^{2}+3 b-6 \)
Ask by Bradley Mcdonald. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The polynomial \( b^{3}-2b^{2}+3b-6 \) factors to \( (b-2)(b^{2}+3) \).
Solution
Factor the expression \( b^{3}-2b^{2}+3b-6 \).
Factor the expression by following steps:
- step0: Factor:
\(b^{3}-2b^{2}+3b-6\)
- step1: Rewrite the expression:
\(b^{3}+3b-2b^{2}-6\)
- step2: Factor the expression:
\(b\left(b^{2}+3\right)-2\left(b^{2}+3\right)\)
- step3: Factor the expression:
\(\left(b-2\right)\left(b^{2}+3\right)\)
The factored form of the polynomial \(b^{3}-2b^{2}+3b-6\) is \((b-2)(b^{2}+3)\).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To factor the polynomial \( b^{3}-2 b^{2}+3 b-6 \), we can apply the Rational Root Theorem. Testing possible rational roots, we find that \( b=2 \) works. Performing synthetic division with \( b-2 \), we reduce the polynomial to \( (b-2)(b^{2}+3) \). The result indicates that our polynomial factors into \( (b-2)(b^{2}+3) \). However, since \( b^2 + 3 \) doesn't factor further with real numbers, we can also express the factorization over the complex numbers as \( (b-2)(b+i\sqrt{3})(b-i\sqrt{3}) \). Voilà! And there you have it!
