x^4+4

=(x^2)^2+4x^2+4-4x^2

=(x^2+2)^2-4x^2

=(x^2-2x+2)(x^2+2x+2)

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

x⁵ + x⁴ + 1 = x⁵ - x³ - x² - x⁴ + x² + x + x³ - x - 1

= x² ( x³ - x - 1 ) - x ( x³ - x - 1 ) + 1 ( x³ - x - 1 )

= ( x³ - x - 1 ) ( x² - x + 1 )

x⁵+x⁴+1

= x⁵+x²-x²+x⁴-x+x+1

=x²(x³-1) + (x³+1) +x²+x+1

= x²(x-1)(x²+x+1)+x(x-1)( x²+x+1) +x²+x+1

=( x² + x+1)( x³-x²+x²-x+1)

=(x² + x+1)( x³-x+1)

      \(54x^3+16y^3\)

\(=2\left(27x^3+8y^3\right)\)

\(=2\left[\left(3x\right)^3+\left(2y\right)^3\right]\)

\(=2\left(3x+2y\right)\left(9x^2-6xy+4y^2\right)\)

       \(x^4-16y^4\)

\(=\left(x^2\right)^2-\left(4y^2\right)^2\)

\(=\left(x^2-4y^2\right)\left(x^2+4y^2\right)\)

\(=\left(x-2y\right)\left(x+2y\right)\left(x^2+4y^2\right)\)

Chúc bạn học tốt.

\(54x^3+16y^3=2\left(27x^3+8y^3\right)\)

\(=2\left[\left(3x\right)^3+\left(2y\right)^3\right]\)

\(=2\left(3x+2y\right)\left[\left(3x\right)^2-3x.2y+\left(2y\right)^2\right]\)

\(=2\left(3x+2y\right)\left(9x^2-6xy+4y^2\right)\)

a, 4x⁴+1 

=(2x)²+1

=(2x+1)(2x-1)

b,c tách làm bình phương rồi làm tương tự

Ko có hằng đẳng thức đó bạn ơi../

\(x^4+5x^3+10x-4\)

\(=x^4+5x^3-2x^2+2x^2+10x-4\)

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

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

Mình cũng vừa làm được cách 2:

\(x^4+5x^3+10x-4\)

=\(x^4-4+5x^3+10x\)

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

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

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

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

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

\(x^4-5x^2+4\)

\(\Leftrightarrow x^4-x^2-4x^2+4\)

\(\Leftrightarrow\left(x^2-1\right).\left(x^2+1\right)-4\left(x^2-1\right)\)

\(\Leftrightarrow\left(x^2-1\right).\left(x^2+1-4\right)\)

\(\Leftrightarrow\left(x^2-1\right).\left(x^2-3\right)\)

Chúc bạn học tốt !

a) \(x^{5} - x^{4} - 2 x^{3} + 2 x^{2} + x - 1\)
Nhóm các hạng tử:

\(\left(\right. x^{5} - x^{4} \left.\right) + \left(\right. - 2 x^{3} + 2 x^{2} \left.\right) + \left(\right. x - 1 \left.\right) = \left(\right. x - 1 \left.\right) \left(\right. x^{4} - 2 x^{2} + 1 \left.\right) .\)

Đặt \(t = x^{2}\) thì \(x^{4} - 2 x^{2} + 1 = \left(\right. t - 1 \left.\right)^{2} = \left(\right. x^{2} - 1 \left.\right)^{2}\).
Vậy

\(\boxed{x^{5} - x^{4} - 2 x^{3} + 2 x^{2} + x - 1 = \left(\right. x - 1 \left.\right) \left(\right. x^{2} - 1 \left.\right)^{2} = \left(\right. x - 1 \left.\right)^{3} \left(\right. x + 1 \left.\right)^{2} .}\)

b) \(x^{3} - 5 x^{2} - 14 x\)
Lấy \(x\) chung:

\(x \left(\right. x^{2} - 5 x - 14 \left.\right) = x \left(\right. x - 7 \left.\right) \left(\right. x + 2 \left.\right) .\)

\(\boxed{x^{3} - 5 x^{2} - 14 x = x \left(\right. x - 7 \left.\right) \left(\right. x + 2 \left.\right) .}\)

c) \(2 x^{2} + 2 x y - 4 y^{2}\)
Lấy \(2\) chung: \(2 \left(\right. x^{2} + x y - 2 y^{2} \left.\right)\).
Nhân tử hóa: \(x^{2} + x y - 2 y^{2} = \left(\right. x + 2 y \left.\right) \left(\right. x - y \left.\right)\).
\(\boxed{2 x^{2} + 2 x y - 4 y^{2} = 2 \left(\right. x + 2 y \left.\right) \left(\right. x - y \left.\right) .}\)

d) \(3 x^{2} + 8 x y - 3 y^{2}\)
Thử phân tích:

\(3 x^{2} + 8 x y - 3 y^{2} = \left(\right. 3 x - y \left.\right) \left(\right. x + 3 y \left.\right) .\)

\(\boxed{3 x^{2} + 8 x y - 3 y^{2} = \left(\right. 3 x - y \left.\right) \left(\right. x + 3 y \left.\right) .}\)

e) \(x^{2} - x - x y - 2 y^{2} + 2 y\)
Gộp lại theo \(x\): \(x^{2} + x \left(\right. - 1 - y \left.\right) + \left(\right. - 2 y^{2} + 2 y \left.\right)\).
Định thức là một bình phương → nghiệm \(x = 2 y\) và \(x = 1 - y\).
Vậy

\(\boxed{x^{2} - x - x y - 2 y^{2} + 2 y = \left(\right. x - 2 y \left.\right) \left(\right. x + y - 1 \left.\right) .}\)

f) \(x^{2} + 2 y^{2} - 3 x y + x - 2 y\)
Xem như phương trình bậc hai theo \(x\): nghiệm \(x = 2 y\) và \(x = y - 1\).
Do đó

\(\boxed{x^{2} + 2 y^{2} - 3 x y + x - 2 y = \left(\right. x - 2 y \left.\right) \left(\right. x - y + 1 \left.\right) .}\)

a: \(x^5-x^4-2x^3+2x^2+x-1\)

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

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

\(=\left(x-1\right)\left(x^2-1\right)^2=\left(x-1\right)\cdot\left(x-1\right)^2\cdot\left(x+1\right)^2\)

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

b: \(x^3-5x^2-14x\)

\(=x\left(x^2-5x-14\right)\)

\(=x\left(x^2-7x+2x-14\right)\)

=x[x(x-7)+2(x-7)]

=x(x-7)(x+2)

c: \(2x^2+2xy-4y^2\)

\(=2\left(x^2+xy-2y^2\right)\)

\(=2\left(x^2+2xy-xy-2y^2\right)\)

=2[x(x+2y)-y(x+2y)]

=2(x+2y)(x-y)

d: \(3x^2+8xy-3y^2\)

\(=3x^2+9xy-xy-3y^2\)

=3x(x+3y)-y(x+3y)

=(x+3y)(3x-y)

e: \(x^2-x-xy-2y^2+2y\)

\(=\left(x^2-xy-2y^2\right)-\left(x-2y\right)\)

\(=\left(x^2-2xy+xy-2y^2\right)-\left(x-2y\right)\)

=x(x-2y)+y(x-2y)-(x-2y)

=(x-2y)(x+y-1)

f: \(x^2+2y^2-3xy+x-2y\)

\(=x^2-2xy-xy+2y^2+x-2y\)

=x(x-2y)-y(x-2y)+(x-2y)

=(x-2y)(x-y+1)

f) \(x^2-6x+5=\left(x^2-x\right)+\left(-5x+5\right)=x\left(x-1\right)-5\left(x-1\right)=\left(x-1\right)\left(x-5\right)\)

g) \(x^4+64=\left(x^2+4x+8\right)\left(x^2-4x+8\right)\)

\(x^2-6x+5\)

\(=\left(x^2-2.3x+3^2\right)-4\)

\(=\left(x-3\right)^2-2^2\)

\(=\left(x-3-2\right)\left(x-3+2\right)\)

\(=\left(x-5\right)\left(x-1\right)\)

(x-1)(x+2)²

x⁴-25x²+26x-4

= (x⁴-25x²)+ (26x-4)

= ((x²)²-(5x)²)+ 2(13x-2)

= (x²-5x)(x²+5x)