Question

Using factor theorem, factorize the polynomiaql \(x^{3}-6x^{2}+11x-6.\). Write the factors.

Answer 1

Let \(f(x)=x^{3}-6x^{2}+11x-6\)
The constant term in \(f(x)\) is equal to \(-6\) and factors of \(-6\) are \(\pm 1,\pm 2,\pm 3,\pm 6.\)
Putting \(x=1\) in \(f(x)\), we have
\(f(1)=1^{3}-6\times 1^{2}+11\times1-6\)
\(=1-6+11-6=0\)
\(\therefore (x-1)\) is a factor of \(f(x)\)
Similarly, \(x-2\) and \(x-3\) are factors of \(f(x).\)
Since \(f(x)\) is a polynomial of degree \(3\). So, it can not have more than three linear factors.
Let \(f(x)=k(x-1)(x-2)(x-3).\) Then,
\(x^{3}-6x^{2}+11x-6=k(x-1)(x-2)(x-3)\)
Putting \(x=0\) on both sides, we get
\(-6=k(0-1)(0-2)(0-3)\)
\(\Rightarrow -6=-6k\Rightarrow k=1\)
Putting \(k=1\) in \(f(x)=k(x-1)(x-2)(x-3),\) we get
\(f(x)=(x-1)(x-2)(x-3)\)
Hence, \(x^{3}-6x^{2}+11x-6=(x-1)(x-2)(x-3)\)