Find all the real zeros of the polynomial. P(x)=3x^{3}+5x^{2}-2x-4

geduiwelh 2021-08-12 Answered
Find all the real zeros of the polynomial.
\(\displaystyle{P}{\left({x}\right)}={3}{x}^{{{3}}}+{5}{x}^{{{2}}}-{2}{x}-{4}\)

Expert Community at Your Service

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

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

i1ziZ
Answered 2021-08-13 Author has 4811 answers
For the zeros:
Put \(\displaystyle{P}{\left({x}\right)}={0}\)
So,
\(\displaystyle{P}{\left({x}\right)}={0}\)
\(\displaystyle\Rightarrow{3}{x}^{{{3}}}+{5}{x}^{{{2}}}-{2}{x}-{4}={0}\)
\(\displaystyle\Rightarrow{\left({x}+{1}\right)}{\left({3}{x}^{{{2}}}+{2}{x}-{4}\right)}={0}\)
Using the Zero Factor Principle: If \(\displaystyle{a}{b}={0}\) then \(\displaystyle{a}={0}\) or \(\displaystyle{b}={0}\)
So,
\(\displaystyle{x}+{1}={0}\) or \(\displaystyle{3}{x}^{{{2}}}+{2}{x}-{4}={0}\)
Now,
For a quadratic equation of the form \(\displaystyle{a}{x}^{{{2}}}+{b}{x}+{c}={0}\) the solution are
\(\displaystyle{x}={\frac{{-{b}\pm\sqrt{{{b}^{{{2}}}-{4}{a}{c}}}}}{{{2}{a}}}}\)
Therefore,
\(\displaystyle{x}+{1}={0}\Rightarrow{x}=-{1}\)
And
\(\displaystyle{3}{x}^{{{2}}}+{2}{x}-{4}={0}\)
\(\displaystyle\Rightarrow{x}={\frac{{-{2}\pm\sqrt{{{2}^{{{2}}}-{4.3}{\left(-{4}\right)}}}}}{{{2}\cdot{3}}}}\)
\(\displaystyle\Rightarrow{x}={\frac{{-{2}\pm{2}\sqrt{{{13}}}}}{{{2}\cdot{3}}}}\)
\(\displaystyle\Rightarrow{x}={\frac{{-{1}\pm\sqrt{{{13}}}}}{{{3}}}}\)
\(\displaystyle\Rightarrow{x}={\frac{{-{1}+\sqrt{{{13}}}}}{{{3}}}},{\frac{{-{1}-\sqrt{{{13}}}}}{{{3}}}}\)
Hence, the real zeros of the given polynomial are:
\(\displaystyle{x}=-{1},{\frac{{-{1}+\sqrt{{{13}}}}}{{{3}}}},{\frac{{-{1}-\sqrt{{{13}}}}}{{{3}}}}\)
Have a similar question?
Ask An Expert
15
 

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

asked 2021-08-07
To find:
The real zeros of polynomial function form an arithmetic sequaence
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{{4}}}-{4}{x}^{{{3}}}-{4}{x}^{{{2}}}+{16}{x}\)
asked 2021-07-29
Nested Form of a Polynomial Expand Q to prove that the polynomials P and Q are the same.
\(\displaystyle{P}{\left({x}\right)}={3}{x}^{{{4}}}-{5}{x}^{{{3}}}+{x}^{{{2}}}-{3}{x}+{5}\)
\(\displaystyle{Q}{\left({x}\right)}={\left({\left({\left({3}{x}-{5}\right)}{x}+{1}\right)}{x}-{3}\right)}{x}+{5}\)
Try to evalue P(2) and Q(2) in your head, using the forms given. Which is easier? Now write the polinomial \(\displaystyle{R}{\left({x}\right)}={x}^{{{5}}}-{2}{x}^{{{4}}}+{3}{x}^{{{3}}}-{2}{x}^{{{2}}}+{3}{x}+{4}\) in "nested" form, like the polinomial Q. Use the nested form to find R(3) in your head.
asked 2021-08-08
DISCUSS DISCOVER: How Many Real Zeros Can a Polynomial Have? Give examples of polynomials that have the following properties, or explain why it is impossible to find such a polynomial.
(a) A polynomial of degree 3 that has no real zeros
(b) A polynomial of degree 4 that has no real zeros
(c) A polynomial of degree 3 that has three real zeros, only one of which is rational
(d) A polynomial of degree 4 that has four real zeros, none of which is rational
What must be true about the degree of a polynomial with integer coefficients if it has no real zeros?
asked 2021-08-08
Indicate whether the expression defines a polynomial function.
\(\displaystyle{P}{\left({x}\right)}=-{x}^{{{2}}}+{3}{x}+{3}\)
polynomial or not a polynomial
If it is a polynomial function, identify the following. (If it is not a polynomial function, enter DNE for all three answers.)
(a) Identify the leading coefficient.
(b) Identify the constant term.
(c) State the degree.
asked 2021-10-27
DISCUSS DISCOVER: How Many Real Zeros Can a Polynomial Have? Give examples of polynomials that have the following properties, or explain why it is impossible to find such a polynomial.
(a) A polynomial of degree 3 that has no real zeros
(b) A polynomial of degree 4 that has no real zeros
(c) A polynomial of degree 3 that has three real zeros, only one of which is rational
(d) A polynomial of degree 4 that has four real zeros, none of which is rational
What must be true about the degree of a polynomial with integer coefficients if it has no real zeros?
asked 2021-08-14
Find all rational zeros of the polynomial, and write the polynomial in factored form.
\(\displaystyle{P}{\left({x}\right)}={4}{x}^{{{4}}}-{37}{x}^{{{2}}}+{9}\)
asked 2021-04-02
Find all rational zeros of the polynomial, and write the polynomial in factored form.
\(\displaystyle{P}{\left({x}\right)}={2}{x}^{{{3}}}-{3}{x}^{{{2}}}-{2}{x}+{3}\)
...