\(f\left(x\right)=5x^4-x^2\left(x-3\right)+3x\left(x-2\right)-6x+2\)

\(=5x^4-x^3+3x^2+3x^2-6x-6x+2\)

\(=5x^4-x^3+6x^2-12x+2\)

\(g\left(x\right)=2x^2\cdot x^2-4x^2+2\left(x+1\right)+5=2x^4-4x^2+2x+7\)

\(f\left(x\right)+g\left(x\right)=7x^4-x^3+2x^2-10x+9\)

\(f\left(x\right)-g\left(x\right)=3x^4-x^3+10x^2-14x-5\)

Ta có : Nghiệm của g(x) là x = 2 và x = -1

=> Để f(x) chia hết cho g(x) thì f(x) cũng nhận x = 2 và x = -1 làm nghiệm

+) f(2) = 0 < tự thế x để tìm a >

+) f(-1) = 0 < tương tự >

=> a = -30 hoặc a = -9 thì f(x) chia hết cho g(x)

\(f\left(x\right)=-3x^2+x-1+x^4-x^3-x^2+3x^4+2x^3\)

\(f\left(x\right)=\left(x^4+3x^4\right)-\left(x^3-2x^3\right)-\left(3x^2+x^2\right)+x-1\)

\(f\left(x\right)=4x^4+x^3-4x^2+x-1\)

\(g\left(x\right)=x^4+x^2-x^3+x-5+5x^3-x^2-3x^4\)

\(g\left(x\right)=\left(x^4-3x^4\right)+\left(5x^3-x^3\right)+\left(x^2-x^2\right)+x-5\)

\(g\left(x\right)=-2x^4+4x^3+x-5\)

`@` `\text {Ans}`

`\downarrow`

`a,`

\(f(x) -3x^2 + x - 1 + x^4 - x^3 - x^2 + 3x^4 + 2x^3\)

`= (x^4 +3x^4) + (-x^3 +2x^3) + (-3x^2 - x^2) + x - 1`

`= 4x^4 + x^3 -4x^2 + x -1`

\(g(x) = x^4 + x^2 - x^3 + x - 5 + 5x^3 - x^2 - 3x^4\)

`= (x^4-3x^4) + (-x^3+5x^3) + (x^2 - x^2) + x -5`

`= -2x^4 + 4x^3 +x - 5`

f(x) = x⁴ - 9x³ + 21x² + ax + b 

g(x) = x² - x - 2

Ta có f(x) bậc 4 ; g(x) bậc 2

=> Thương là một đa thức bậc 2

Gọi đa thức thương đó là h(x) = x² + cx + d

Ta có f(x) chia hết cho g(x)

<=> x⁴ - 9x³ + 21x² + ax + b = ( x² - x - 2 )( x² + cx + d )

<=> x⁴ - 9x³ + 21x² + ax + b = x⁴ + cx³ + dx² - x³ - cx² - dx - 2x² - 2cx - 2d

<=> x⁴ - 9x³ + 21x² + ax + b = x⁴ + ( c - 1 )x³ + ( d - c - 2 )x² + ( -d - 2c )x - 2d

Đồng nhất hệ số ta được :

\(\hept{\begin{cases}c-1=-9\\d-c-2=21\\-d-2c=a\end{cases}};-2d=b\)

\(\Rightarrow\hept{\begin{cases}c=-8\\d=15\\a=1\end{cases}};b=-30\)

\(\Rightarrow\hept{\begin{cases}a=1\\b=-30\end{cases}}\)

Vậy ...

a) \(\left(x+3\right)^2-\left(x-4\right)\left(x+8\right)=1\)

\(\Leftrightarrow\left(x^2+6x+9\right)-\left(x^2+4x-32\right)-1=0\)

\(\Leftrightarrow2x=-40\)

\(\Rightarrow x=-20\)

b) \(\left(x+3\right)\left(x^2-3x+9\right)-x\left(x-2\right)\left(x+2\right)=15\)

\(\Leftrightarrow x^3+27-x^3+4x=15\)

\(\Leftrightarrow4x=-12\)

\(\Rightarrow x=-3\)

c) \(\left(x-2\right)^2-\left(x+3\right)^2-4\left(x+1\right)=5\)

\(\Leftrightarrow\left(x^2-4x+4\right)-\left(x^2+6x+9\right)-\left(4x+4\right)=5\)

\(\Leftrightarrow-14x=14\)

\(\Rightarrow x=-1\)

d) \(\left(2x-3\right)\left(2x+3\right)-\left(x-1\right)^2-3x\left(x-5\right)=-44\)

\(\Leftrightarrow4x^2-9-\left(x^2-2x+1\right)-\left(3x^2-15x\right)=-44\)

\(\Leftrightarrow17x=-34\)

\(\Rightarrow x=-2\)

e) \(\left(x-2\right)^3-\left(x-3\right)\left(x^2+3x+9\right)+6\left(x+1\right)^2=49\)

\(\Leftrightarrow x^3-6x^2+12x-8-x^3+27+6x^2+12x+6=49\)

\(\Leftrightarrow24x=24\)

\(\Rightarrow x=1\)

f) \(5x\left(x-3\right)^2-5\left(x-1\right)^3+15\left(x+2\right)\left(x-2\right)=5\)

\(\Leftrightarrow5x^3-30x^2+45x-5x^3+15x^2-15x+5+15x^2-60=5\)

\(\Leftrightarrow30x=60\)

\(\Rightarrow x=2\)

g) \(\left(x+3\right)^3-x\left(3x+1\right)^2+\left(2x+1\right)\left(4x^2-2x+1\right)-3x^2=42\)

\(\Leftrightarrow x^3+9x^2+27x+27-9x^3-6x^2-x+8x^3+1-3x^2=42\)

\(\Leftrightarrow26x=14\)

\(\Rightarrow x=\frac{7}{13}\)

b, x = -5/3 hoặc x = 4/3.

c, x = 0 hoặc x = 3, -3.

d, x = 0 hoặc x = 2, -2.

e, x = 1 hoặc x = \(\dfrac{-1}{2}\).

a: \(\Leftrightarrow x^2-40x+400-x^2-4x-3=-7\)

=>-44x+397=-7

=>-44x=-404

hay x=101

b: \(\Leftrightarrow\left[{}\begin{matrix}3x+5=0\\4-3x=0\end{matrix}\right.\Leftrightarrow x\in\left\{-\dfrac{5}{3};\dfrac{4}{3}\right\}\)

c: \(\Leftrightarrow x\left(x^2-9\right)=0\)

=>x(x-3)(x+3)=0

hay \(x\in\left\{0;3;-3\right\}\)

d: \(\Leftrightarrow x\left(x-2\right)\left(x+2\right)=0\)

hay \(x\in\left\{0;2;-2\right\}\)

e: =>(2x+1)(1-x)=0

=>x=-1/2 hoặc x=1