Step 1

Given polynomial is, \(\displaystyle{P}{\left({x}\right)}={6}{x}^{{{4}}}-{23}{x}^{{{3}}}-{13}{x}^{{{2}}}+{32}{x}+{16}\).

The zeroes are found as follows:

\(\displaystyle{6}{x}^{{{4}}}-{23}{x}^{{{3}}}-{13}{x}^{{{2}}}+{32}{x}+{16}\)

\(\displaystyle{6}{x}^{{{3}}}{\left({x}+{1}\right)}-{29}{x}^{{{2}}}{\left({x}+{1}\right)}+{16}{x}{\left({x}+{1}\right)}+{16}{\left({x}+{1}\right)}={0}\)

\(\displaystyle{\left({x}+{1}\right)}{\left({6}{x}^{{{3}}}-{29}{x}^{{{2}}}+{16}{x}+{16}\right)}={0}\)

\(\displaystyle{\left({x}+{1}\right)}{\left({2}{x}+{1}\right)}{\left({3}{x}^{{{2}}}-{16}{x}+{16}\right)}={0}\)

(x+1)(x-4)(2x+1)(3x-4)=0

Step 2

On solving further,

x+1=0 or x-4=0 or 2x+1=0 or 3x-4=0

\(\displaystyle{x}=-{1},{x}={4},{x}=-{\frac{{{1}}}{{{2}}}},{x}={\frac{{{4}}}{{{3}}}}\)

Thus, rational zeroes are \(\displaystyle-{\frac{{{1}}}{{{2}}}}\) and \(\displaystyle{\frac{{{4}}}{{{3}}}}\). And, the factored form of the polynomial is (x+1)(x-4)(2x+1)(3x-4).

Given polynomial is, \(\displaystyle{P}{\left({x}\right)}={6}{x}^{{{4}}}-{23}{x}^{{{3}}}-{13}{x}^{{{2}}}+{32}{x}+{16}\).

The zeroes are found as follows:

\(\displaystyle{6}{x}^{{{4}}}-{23}{x}^{{{3}}}-{13}{x}^{{{2}}}+{32}{x}+{16}\)

\(\displaystyle{6}{x}^{{{3}}}{\left({x}+{1}\right)}-{29}{x}^{{{2}}}{\left({x}+{1}\right)}+{16}{x}{\left({x}+{1}\right)}+{16}{\left({x}+{1}\right)}={0}\)

\(\displaystyle{\left({x}+{1}\right)}{\left({6}{x}^{{{3}}}-{29}{x}^{{{2}}}+{16}{x}+{16}\right)}={0}\)

\(\displaystyle{\left({x}+{1}\right)}{\left({2}{x}+{1}\right)}{\left({3}{x}^{{{2}}}-{16}{x}+{16}\right)}={0}\)

(x+1)(x-4)(2x+1)(3x-4)=0

Step 2

On solving further,

x+1=0 or x-4=0 or 2x+1=0 or 3x-4=0

\(\displaystyle{x}=-{1},{x}={4},{x}=-{\frac{{{1}}}{{{2}}}},{x}={\frac{{{4}}}{{{3}}}}\)

Thus, rational zeroes are \(\displaystyle-{\frac{{{1}}}{{{2}}}}\) and \(\displaystyle{\frac{{{4}}}{{{3}}}}\). And, the factored form of the polynomial is (x+1)(x-4)(2x+1)(3x-4).