Question

On dividing ${{x}^{3}}-3{{x}^{2}}+x+2$ by a polynomial g(x), the quotient and remainder were $x-2$ and $-2x+4$, respectively. Find g(x).
(a) ${{x}^{2}}+x+1$
(b) ${{x}^{2}}-x+1$
(c) ${{x}^{2}}-x-1$
(d) ${{x}^{2}}+1$

Answer 1

Hint: We know the fundamental algorithm of division is given by,
Dividend=Divisor*Quotient + Remainder. This rule can be used even for division of polynomials.

Complete step-by-step answer:
From the question we have the dividend as ${{x}^{3}}-3{{x}^{2}}+x+2$ and the quotient and remainder as $x-2$ and $-2x+4$ . We need to use the fundamental theorem of division to find the divisor which is g(x). Hence, we can write

${{x}^{3}}-3{{x}^{2}}+x+2=g(x)(x-2)+(-2x+4)$

Now subtracting both sides with -2x+4 we have,

${{x}^{3}}-3{{x}^{2}}+x+2-(-2x+4)=g(x)(x-2)$

On simplification we get,


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


Now we can write g(x) as-

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

We can see that the polynomial g(x) is in a form which can be evaluated by division of polynomials. For that we need to divide ${{x}^{3}}-3{{x}^{2}}+3x-2$ by $x-2$ .

Division of polynomials is done in a very similar way by which division of actual numbers is done. Let us do this without pen and paper. We need to be very inspective to do this. To do this we need to ask ourselves on multiplying what with this will we get this.

We need to ask on multiplying what with $x$ can we get ${{x}^{3}}$ and the answer is ${{x}^{2}}$ . Now we will multiply ${{x}^{2}}$ with x-2 and subtract it from ${{x}^{3}}-3{{x}^{2}}+3x-2$ by which we get $-{{x}^{2}}+3x-2$ we will follow the same procedure till on subtracting we don’t get zero.

Now we have $-{{x}^{2}}+3x-2$ . On multiplying x-2 with –x we get $-{{x}^{2}}+2x$ and subtracting this from $-{{x}^{2}}+3x-2$ we get x-2. Now on multiplying x-2 with 1 we get x-2. Hence, the remainder now is zero means our division is completed. The term with which we were multiplying gives us our divisor. Therefore, the divisor is ${{x}^{2}}-x+1$ . Hence, we can write $g(x)={{x}^{2}}-x+1$ .

Hence, (b) is the correct option.

Note: We need to notice one thing that on dividing ${{x}^{3}}-3{{x}^{2}}+3x-2$ with $x-2$ the remainder cannot be anything but 0 because we followed the fundamental definition here and if we get anything other than 0 as remainder then our calculation must have been wrong somewhere.