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)=a₁+10 x+a₂ x^2+a₃ x^3+x^4, & g(x)=b₁+3 x+b₂ x^2+b₃ x^3+x^4, & h(x)=f(x+1)-g(x+2) . aligned If f(x)

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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)=a₁+10 x+a₂ x^2+a₃ x^3+x^4, & g(x)=b₁+3 x+b₂ x^2+b₃ 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

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Correct answer: C. -4

aligned & h(x)=f(x+1)-g(x+2) & =a₁+10(x+1)+a₂(x+1)^2+a₃(x+1)^3+(x+1)^4-b₁-3(x+2)-b₂(x+2)^2-b₃(x+2)^3-(x+2)^4 aligned Coeff. of x^3 in h(x) aligned & =a₃-b₃-4 & f(x)-g(x) 0 x R & a₁+10 x+a₂ x^2+a₃ x^3+x^4-b₁-3 x-b₂ x^2-b₃ x^3-x^4 0 & x^3 (a₃-b₃ )+x^2 (a₂-b₂ )+7 x+ (a₁-b₁ ) 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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