Question

Show that p(x) = x3 – 3x2 + 2x – 6 has only one real zero.

Answer 1

So, hence there is only one zero.

Answer 2

Answer:

$p(x)=x^{3} -3x^{2} +2x-6$ has only one zero.

Step-by-step explanation:

Given: $p(x)=x^{3} -3x^{2} +2x-6$.

To show: p(x) has only one real zero.

Recall: If $x=a$ is zero of polynomial then $p(a)=0$.

Factorize p(x).

$p(x)=x^{3} -3x^{2} +2x-6$

       $=x^{2} (x-3)+2(x-3)\\=(x^{2} +2)(x-3)$

For zero of polynomial $p(x)=0$ which is possible only if $x-3=0$ or $x^{2} +2=0$ but $x^{2} +2=0$ gives imaginary roots.

So, the only possible zero is $x=3.$

Hence proved.

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