Question

If $(x-1)$ is one of the factors of the polynomial $(x^3-5x^2+8x-4),$ then find the other factors using synthetic division.

• A. $(x+1{)}^2$
• B. $(x-3{)}^2$
• C. $(x-2)(x-3)$
• D. $(x-2{)}^2$

Answer 1

The correct option is D $(x-2{)}^2$

$p\left(x\right)=(x^3-5x^2+8x-4)$
One factor = $(x-1)$

Using synthetic division,
$\begin{array}{ccccc}& x^3& x^2& x^1& x^0\\ & 1& -5& 8& -4\\ 1& \downarrow & 1& -4& 4\\ & 1& -4& 4& |-0\end{array}$

$\therefore p\left(x\right)=(x-1)(x^2-4x+4)\text{Now,}{x}^2-4x+4=x^2-2x-2x+4=x(x-2)-2(x-2)=(x-2)(x-2)\text{Hence,}{x}^3-5x^2+8x-4=(x-1)(x-2)(x-2)=(x-1)(x-2{)}^2$