Question

Factor each polynomial. (b+3)^{3}-27

Polynomial factorization
ANSWERED
asked 2021-04-24
Factor each polynomial.
\(\displaystyle{\left({b}+{3}\right)}^{{{3}}}-{27}\)

Answers (1)

2021-04-26
Step 1
Given
\(\displaystyle{\left({b}+{3}\right)}^{{{3}}}-{27}={\left({b}+{3}\right)}^{{{3}}}-{\left({3}\right)}^{{{3}}}\)
Step 2
\(\displaystyle\because{a}^{{{3}}}-{b}^{{{3}}}={\left({a}+{b}\right)}{\left({a}^{{{2}}}+{a}{b}+{b}^{{{2}}}\right)}\)
\(\displaystyle\Rightarrow{\left({b}+{3}\right)}^{{{3}}}-{\left({3}\right)}^{{{3}}}={\left({b}+{3}-{3}\right)}{\left[{\left({b}+{3}\right)}^{{{2}}}+{\left({b}+{3}\right)}{\left({3}\right)}+{\left({3}\right)}^{{{2}}}\right]}\)
\(\displaystyle={\left({b}\right)}{\left[{b}^{{{2}}}+{6}{b}+{9}+{3}{b}+{9}+{9}\right]}\)
\(\displaystyle={b}{\left[{b}^{{{2}}}+{9}{b}+{27}\right]}\)
\(\displaystyle{\left({b}+{3}\right)}^{{{3}}}-{27}={b}{\left({b}^{{{2}}}+{9}{b}+{27}\right)}\)
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