Mathematics - Functions Question with Solution | TestHub

$\begin{array}{rl}& h(x)=f(x+1)-g(x+2)\\ & =a_1+10(x+1)+a_2(x+1{)}^2+a_3(x+1{)}^3+(x+1{)}^4-b_1-3(x+2)-b_2(x+2{)}^2-b_3(x+2{)}^3-(x+2{)}^4\end{array}$ Coeff. of $x^3$ in $h(x)$ $\begin{array}{rl}& =a_3-b_3-4\\ & f(x)-g(x)\ne 0\forall x\in R\\ & \Rightarrow a_1+10x+a_2{x}^2+a_3{x}^3+x^4-b_1-3x-b_2{x}^2-b_3{x}^3-x^4\ne 0\\ & \Rightarrow x^3\left(a_3-b_3\right)+x^2\left(a_2-b_2\right)+7x+\left(a_1-b_1\right)\ne 0\end{array}$ Cubic Eq. will become zero at atleast are value of $x$ So it will be quadratic $\Rightarrow {\mathrm{a}}_3-{\mathrm{b}}_3=0$