Let R denote the set of all real numbers. Let a_ i , b_ i R for i \ 1,2,3\ . Define the functions f: R R , g: R R , and h: R R by aligned & f(x)=a_1+10 x+a_2 x^2+a_3 x^3+x^4, \\ & g(x)=b_1+3 x+b_2 x^2+b_3 x^3+x^4, \\ & h(x)=f(x+1)-g(x+2) . aligned If f(x) g(x) for every x R , then the coefficient of x^3 in h(x) is
C. -4
aligned & 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 aligned Coeff. of x^3 in h(x) aligned & =a_3-b_3-4 \\ & f(x)-g(x) 0 x R \\ & a_1+10 x+a_2 x^2+a_3 x^3+x^4-b_1-3 x-b_2 x^2-b_3 x^3-x^4 0 \\ & x^3 (a_3-b_3 )+x^2 (a_2-b_2 )+7 x+ (a_1-b_1 ) 0 aligned Cubic Eq. will become zero at atleast are value of x So it will be quadratic a _3- b _3=0
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