A polynomial and one of its factors is given. Factor the polynomial COMPLETELY given that one of its factors is x+4. Your final answer should be in factored form. f(x)=x^{7}+4x^{6}+7x^{4}+28x^{3}-8x-32

nicekikah 2021-08-05 Answered
A polynomial and one of its factors is given. Factor the polynomial COMPLETELY given that one of its factors is \(\displaystyle{x}+{4}\). Your final answer should be in factored form.
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{{7}}}+{4}{x}^{{{6}}}+{7}{x}^{{{4}}}+{28}{x}^{{{3}}}-{8}{x}-{32}\)

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

Lacey-May Snyder
Answered 2021-08-06 Author has 25559 answers

Step 1
We use the synthetic division to get the other factor for the missing terms, we will use 0 as the coefficient.
\(\begin{array}{|c|c|} \hline -4 & 1 & 4 & 0 & 7 & 28 & 0 & -8 & -32 \\ \hline & & -4 & 0 & 0 & -28 & 0 & 0 & 32 \\ \hline & 1 & 0 & 0 & 7 & 0 & 0 & -8 & 0\\ \hline \end{array}\)
So the quotient is \(\displaystyle{x}^{{{6}}}+{7}{x}^{{{3}}}-{8}\)
Step 2
Then we try to factor \(\displaystyle{x}^{{{6}}}+{7}{x}^{{{3}}}-{8}\)
\(\displaystyle{x}^{{{6}}}+{7}{x}^{{{3}}}-{8}\)
\(\displaystyle={\left({x}^{{{3}}}+{8}\right)}{\left({x}^{{{3}}}-{1}\right)}\)
\(\displaystyle={\left({x}^{{{3}}}+{2}^{{{3}}}\right)}{\left({x}^{{{3}}}-{1}^{{{3}}}\right)}\)
\(\displaystyle={\left({x}+{2}\right)}{\left({x}^{{{2}}}-{2}{x}+{4}\right)}{\left({x}-{1}\right)}{\left({x}^{{{2}}}+{x}+{1}\right)}\)
Answer: \(\displaystyle{\left({x}+{4}\right)}{\left({x}+{2}\right)}{\left({x}^{{{2}}}-{2}{x}+{4}\right)}{\left({x}-{1}\right)}{\left({x}^{{{2}}}+{x}+{1}\right)}\)

Not exactly what you’re looking for?
Ask My Question
25
 

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 2021-08-06
Given \(\displaystyle{P}{\left({x}\right)}={3}{x}^{{{5}}}-{4}{x}^{{{4}}}+{9}{x}^{{{3}}}-{12}{x}^{{{2}}}-{12}{x}+{16}\), and that 2i is a zero, write P in factored form (as a product of linear factors). Be sure to write the full equation, including \(\displaystyle{P}{\left({x}\right)}=\).
asked 2021-08-02
Given \(\displaystyle{P}{\left({x}\right)}={3}{x}^{{{5}}}-{x}^{{{4}}}+{81}{x}^{{{3}}}-{27}{x}^{{{2}}}-{972}{x}+{324}\), and that 6i is a zero, write P in factored form (as a product of linear factors). Be sure to write the full equation, including \(\displaystyle{P}{\left({x}\right)}=\).
asked 2021-08-11
Given \(\displaystyle{P}{\left({x}\right)}={3}{x}^{{{5}}}−{5}{x}^{{{4}}}+{37}{x}^{{{3}}}−{83}{x}^{{{2}}}−{176}{x}−{48}\), and that 4i is a zero, write P in factored form (as a product of linear factors). Be sure to write the full equation, including \(\displaystyle{P}{\left({x}\right)}=\).
asked 2021-08-07
Given \(\displaystyle{P}{\left({x}\right)}={3}{x}^{{{5}}}-{8}{x}^{{{4}}}+{35}{x}^{{{3}}}-{98}{x}^{{{2}}}-{208}{x}+{480}\), and that 4i is a zero, write P in factored form (as a product of linear factors). Be sure to write the full equation, including \(\displaystyle{P}{\left({x}\right)}=\)
asked 2021-05-16
Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
asked 2021-08-03
Perform the indicated operation and simplify the result. Leave your answer in factored form.
\(\displaystyle{\left[{\frac{{{\left({4}{x}-{8}\right)}}}{{{\left(-{3}{x}\right)}}}}\right]}.{\left[{\frac{{{12}}}{{{\left({12}-{6}{x}\right)}}}}\right]}\)
asked 2021-03-03
Find all rational zeros of the polynomial, and write the polynomial in factored form.
\(\displaystyle{P}{\left({x}\right)}={2}{x}^{{{3}}}+{7}{x}^{{{2}}}+{4}{x}-{4}\)
...