Question

Find the binomial that completes the factorization. 5s^4+5000st^3= 5s(s^3+1000t^3)

Polynomial factorization
ANSWERED
asked 2020-11-24
Find the binomial that completes the factorization.
\(\displaystyle{5}{s}^{{4}}+{5000}{s}{t}^{{3}}={5}{s}{\left({s}^{{3}}+{1000}{t}^{{3}}\right)}\)

Answers (1)

2020-11-25
Step 1
Given:
\(\displaystyle{5}{s}^{{4}}+{5000}{s}{t}^{{3}}={5}{s}{\left({s}^{{3}}+{1000}{t}^{{3}}\right)}\)
We know that
\(\displaystyle{a}^{{3}}+{b}^{{3}}={\left({a}+{b}\right)}{\left({a}^{{2}}+{a}{b}+{b}^{{2}}\right)}\)
So,
\(\displaystyle{\left({s}^{{3}}+{1000}{t}^{{3}}\right)}={\left({s}^{{3}}+{\left({10}{t}\right)}^{{3}}\right)}\)
Step 2
\(\displaystyle{\left({s}^{{3}}+{\left({10}{t}\right)}^{{3}}\right)}={\left({s}+{10}{t}\right)}{\left({s}^{{2}}-{\left({s}\right)}{\left({10}{t}\right)}+{\left({10}{t}\right)}^{{2}}\right)}\)
\(\displaystyle={\left({s}+{10}{t}\right)}{\left({s}^{{2}}-{10}{s}{t}+{10}^{{2}}{t}^{{2}}\right)}\)
Hence
\(\displaystyle{5}{s}^{{4}}+{5000}{s}{t}^{{3}}={\left({s}={10}{t}\right)}{\left({s}^{{2}}-{10}{s}{t}+{10}^{{2}}{t}^{{2}}\right)}\)
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