Question

# Find all zeros of the polynomial P(x)=x^{3}-3x^{2}-10x+24 knowing that x = 2 is a zero of the polynomial.

Applications of integrals

Find all zeros of the polynomial $$\displaystyle{P}{\left({x}\right)}={x}^{{{3}}}-{3}{x}^{{{2}}}-{10}{x}+{24}$$ knowing that $$x = 2$$ is a zero of the polynomial.

2021-05-20

Step 1
The given polynomial is $$\displaystyle{P}{\left({x}\right)}={x}^{{{3}}}-{3}{x}^{{{2}}}-{10}{x}+{24}$$ whose one of the factors is $$x=2$$
Divide the polynomial by $$x-2$$.
Step 2
The quotient is $$\displaystyle{x}^{{{2}}}-{x}-{12}$$.
Factorize the polynomial $$\displaystyle{x}^{{{2}}}-{x}-{12}$$ and find the zeros
$$\displaystyle{x}^{{{2}}}-{x}-{12}={0}$$
$$(x-4)(x+3)=0$$
$$x=4. x=-3$$
Hence, the zeros are $$x=4. x=-3\ and\ x=2.$$