Cho f(x)=ax ³+bx ²+cx+d f(5) – f(4)= 2019 và a ∈ N* Cm: f(7) = f(2) là hợp số 21/08/2021 Bởi Eliza Cho f(x)=ax ³+bx ²+cx+d f(5) – f(4)= 2019 và a ∈ N* Cm: f(7) = f(2) là hợp số
f(5)−f(4)=2019f(5)−f(4)=2019 (a.5^3 + b.5^2 + 5c + d)-(a.2^3 + b.2^2 + 2c + d)=2019$ →61a+9b+c=2019→61a+9b+c=2019 Có: f(7)−f(2)=(a.73+b.72+7c+d)−(a.23+b.22+2c+d)f(7)−f(2)=(a.73+b.72+7c+d)−(a.23+b.22+2c+d) =335a+45+5c=335a+45+5c =30a+5.(61a+9b+c)=30a+5.(61a+9b+c) =30a+5.2019=30a+5.2019 Vì 30a ⋮ 3;2019 ⋮ 3→f(7)−f(2)=30a+5.2019 ⋮ 330a ⋮ 3;2019 ⋮ 3→f(7)−f(2)=30a+5.2019 ⋮ 3 Mà a∈N*→a>0→f(7)−f(2)=30a+5.2019>5.2019>3→a>0→f(7)−f(2)=30a+5.2019>5.2019>3 Do đó f(7)-f(2) là hợp số Bình luận
f(5)−f(4)=2019f(5)−f(4)=2019
(a.5^3 + b.5^2 + 5c + d)-(a.2^3 + b.2^2 + 2c + d)=2019$
→61a+9b+c=2019→61a+9b+c=2019
Có: f(7)−f(2)=(a.73+b.72+7c+d)−(a.23+b.22+2c+d)f(7)−f(2)=(a.73+b.72+7c+d)−(a.23+b.22+2c+d)
=335a+45+5c=335a+45+5c
=30a+5.(61a+9b+c)=30a+5.(61a+9b+c)
=30a+5.2019=30a+5.2019
Vì 30a ⋮ 3;2019 ⋮ 3→f(7)−f(2)=30a+5.2019 ⋮ 330a ⋮ 3;2019 ⋮ 3→f(7)−f(2)=30a+5.2019 ⋮ 3
Mà a∈N*→a>0→f(7)−f(2)=30a+5.2019>5.2019>3→a>0→f(7)−f(2)=30a+5.2019>5.2019>3
Do đó f(7)-f(2) là hợp số