Giải thích các bước giải:
a)
f(5)=$\text{125a}$ + $\text{25b}$ + $\text{5c}$ + d
f(4)=$\text{64a}$ + $\text{16b}$ + $\text{4c}$ + d
f(7)=$\text{343a}$ + $\text{49b}$ + $\text{7c}$ + d
f(2)=$\text{8}$ + $\text{4b}$ + $\text{2c}$ + d
Xét :
f(5)−f(4)=$\text{125a}$ + $\text{25b}$ + $\text{5c}$ + d - $\text{64a}$ + $\text{16b}$ + $\text{4c}$ + d
=$\text{125a}$ + $\text{25b}$ + $\text{5c}$ + d - $\text{64a}$ - $\text{16b}$ - $\text{4c}$- d ( Phá ngoặc đổi dấu )
=$\text{61a}$ + $\text{9b}$ + c = $\text{2020}$
Khi đó :
f(7)- f(2) =$\text{343a}$ + $\text{49b}$ + $\text{7c}$ + d - $\text{8}$ + $\text{4b}$ + $\text{2c}$ + d
=$\text{343a}$ + $\text{49b}$ + $\text{7c}$ + d - $\text{8}$ - $\text{4b}$ - $\text{2c}$ - d (Phá ngoặc đổi dấu )
=$\text{345a}$ + $\text{45b}$ + $\text{5c}$ =$\text{5.(61a+9b+c)+30=5.2020+30$\vdots$5}$
$\Rightarrow$ f(7) -f(2) $\vdots$ 5 $\Rightarrow$ Hợp số
b)
$\text{Ta có : f(2) = 4a + 2b + c}$
$\text{f(-5) = 25a - 5b + c }$
=> f(2) + f(-5) = (4a + 25a) + (2b - 5b) + (c + c) = (29a + 2c) - 3b = 3b - 3b = 0 (Vì 29a + 2c = 3b)
=> f(2) = -f(5)
=> 4a + 2b + c = -(25a - 5b + c)
=> f(2).f(-5) = (4a + 2b + c).(25a + 5b + c) = -(25a + 5b + c)2 < 0 (đpcm)
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