Find all rational zeros of the polynomial, and write the polynomial in factored form. P(x)=2x^{3}-3x^{2}-2x+3

Caelan 2021-04-02 Answered
Find all rational zeros of the polynomial, and write the polynomial in factored form.
\(\displaystyle{P}{\left({x}\right)}={2}{x}^{{{3}}}-{3}{x}^{{{2}}}-{2}{x}+{3}\)

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

Arham Warner
Answered 2021-04-04 Author has 24173 answers
Step 1
\(\displaystyle{P}{\left({x}\right)}={2}{x}^{{{3}}}-{3}{x}^{{{2}}}-{2}{x}+{3}\)
All rational zeros = ?
Factored form = ?
\(\displaystyle{P}{\left({x}\right)}={x}^{{{2}}}{\left({2}{x}-{3}\right)}-{2}{\left({2}{x}-{3}\right)}\)
\(\displaystyle{P}{\left({x}\right)}={\left({2}{x}-{3}\right)}{\left({x}^{{{2}}}-{2}\right)}\)
\(\displaystyle{P}{\left({x}\right)}={\left({2}{x}-{3}\right)}{\left({x}-\sqrt{{{2}}}\right)}{\left({x}+\sqrt{{{2}}}\right)}\)
The polynomial in factored form:
\(\displaystyle{P}{\left({x}\right)}={\left({2}{x}-{3}\right)}{\left({x}-\sqrt{{{2}}}\right)}{\left({x}+\sqrt{{{2}}}\right)}\)
Step 2
To find rational zeros:
Put P(x) = 0
\(\displaystyle{\left({2}{x}-{3}\right)}{\left({x}-\sqrt{{{2}}}\right)}{\left({x}+\sqrt{{{2}}}\right)}={0}\)
\(\displaystyle{2}{x}-{3}={0}\Rightarrow{x}={\frac{{{3}}}{{{2}}}}\)
\(\displaystyle{x}-\sqrt{{{2}}}={0}\Rightarrow{x}=\sqrt{{{2}}}\)
\(\displaystyle{x}+\sqrt{{{2}}}={0}\Rightarrow{x}=-\sqrt{{{2}}}\)
The rational zeros are: \(\displaystyle-\sqrt{{{2}}},\sqrt{{{2}}}\) and \(\displaystyle{\frac{{{3}}}{{{2}}}}\).
Not exactly what you’re looking for?
Ask My Question
40
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

...