Toán Lớp 7: Cho f(x) + g(x) = 6x^4 – 3x^2 -5 ; f(x) – g(x) = 4x^4 – 6x^3 + 7x^2 + 8x – 9 . Hãy tìm các đa thức f(x) ; g(x)

Toán Lớp 7: Cho f(x) + g(x) = 6x^4 – 3x^2 -5 ; f(x) – g(x) = 4x^4 – 6x^3 + 7x^2 + 8x – 9 .
Hãy tìm các đa thức f(x) ; g(x)

TRẢ LỜI

1. maianhnhiquynh
Ta có:
[f(x) + g(x)] – [f(x) – g(x)]
= f(x) + g(x) – f(x) + g(x)
= 2g(x)
=> 2g(x) = [f(x) + g(x)] – [f(x) – g(x)] = (6x^4 – 3x^2 – 5) – (4x^4 – 6x^3 + 7x^2 + 8x – 9)
= 6x^4 – 3x^2 – 5 – 4x^4 + 6x^3 – 7x^2 – 8x + 9
= (6x^4 – 4x^4) + 6x^3 – (3x^2 + 7x^2) – 8x + (-5 + 9)
= 2x^4 + 6x^3 – 10x^2 – 8x + 4
=> g(x) = 1/2(2x^4 + 6x^3 – 10x^2 – 8x + 4)
= x^4 + 3x^3 – 5x^2 – 4x + 2
Lại có:
[f(x) + g(x)] – g(x)
= f(x) + g(x) – g(x)
= f(x)
=> f(x) = [f(x) + g(x)] – g(x) = (6x^4 – 3x^2 – 5) – (x^4 + 3x^3 – 5x^2 – 4x + 2)
= 6x^4 – 3x^2 – 5 – x^4 – 3x^3 + 5x^2 + 4x – 2
= (6x^4 – x^4) – 3x^3 + (-3x^2 + 5x^2) + 4x – (5 + 2)
= 5x^4 – 3x^3 + 2x^2 + 4x – 7
Vậy f(x) = 5x^4 – 3x^3 + 2x^2 + 4x – 7; g(x) = x^4 + 3x^3 – 5x^2 – 4x + 2

Trả lời
2. hienthucthu97
f(x) = [(6x^4 – 3x^2 – 5) + ( 4x^4 – 6x^3 + 7x^2 + 8x -9)]:2
f(x) = (6x^4 – 3x^2 -5 + 4x^4 – 6x^3 + 7x^2 + 8x -9) :2
f(x) = [ (6x^4 + 4x^4) – (3x^2 – 7x^2) – 6x^3  – (5 +9) +8x]:2
f(x) = ( 10x^4 + 4x^2 – 6x^3  – 14 + 8x) :2
f(x)= 5x^4 + 2x^2 – 3x^3 – 7 + 4x
Mà f(x) + g(x) = 6x^4 – 3x^2 – 5
=> g(x) = (6x^4 – 3x^2 -5 )- f(x)
=> g(x) = (6x^4 – 3x^2 – 5) – ( 5x^4 + 2x^2 – 3x^3 – 7 + 4x)
=> g(x) = 6x^4 – 3x^2 – 5 – 5x^4 – 2x^2 + 3x^3 + 7 -4x
=> g(x) = (6x^4 – 5x^4) – (3x^2 + 2x^2) – (5 -7) + 3x^3 – 4x
=> g(x) = x^4 – 5x^2 + 2 + 3x^3 -4x
Vậy f(x) = 5x^4 + 2x^2 – 3x^3 – 7 + 4x ; g(x) = x^4 – 5x^2 + 2 + 3x^3 -4x

Trả lời