Question

Apply the division algorithm to find the quotient and remainder on dividing $f\left(x\right)$ by $g\left(x\right)$.

$f\left(x\right)=x^3-3x^2+5x-3,g\left(x\right)=x^2-2$ [4 MARKS]

Answer 1

Concept : 1 Mark
Application : 1 Mark
Calculation : 2 Marks

We have,

$f\left(x\right)=x^3-3x^2+5x-3$ and $g\left(x\right)=x^2-2$

Here, degree $\left(\right(f\left(x\right))=3$ and degree $\left(g\right(x\left)\right)=2$

Therefore, quotient $q\left(x\right)$ is of degree 1 and the remainder $r\left(x\right)$ is of degree less than 2.

Let $q\left(x\right)=ax+b$ and $r\left(x\right)=cx+d$

Using division algorithm, we have

$f\left(x\right)=g\left(x\right)\times q\left(x\right)+r\left(x\right)$

$\Rightarrow x^3-3x^2+5x-3=(x^2-2)(ax+b)+(cx+d)$

$\Rightarrow x^3-3x^2+5x-3=a{x}^3+b{x}^2+(c-2a)x-2b+d$

Equating the coefficients of various powers of x on both sides, we get

1 = a [On equating the coefficients of $x^3$]

-3 = b [On equating the coefficients of $x^2$]

5 = c - 2a

-3 = - 2b + d

Solving these equations, we get

a = 1, b = -3, c = 7 and d = -9

$\therefore$ Quotient $q\left(x\right)=ax+b=x-3$ and remainder $r\left(x\right)=7x-9$