Phân tích đa thức thành nhân tử:
a) a4 + a2 + 1
= a4 + 2a2 + 1 - a2
= (a2 + 1)2 - a2
= (a2 + 1 - a)(a2 + 1 + a)
b) a4 + a2 - 2
= a4 + 2a2 - a2 - 2
= a2(a2 + 2) - (a2 + 2)
= (a2 + 2)(a2 - 1)
= (a2 + 2)(a - 1)(a + 1)
c) x2 + 4x2 - 5
= 5x2 - 5
= 5(x2 - 1)
= 5(x - 1)(x + 1)
d) x3 - 19x - 30
= x3 - 25x + 6x - 30
= x(x2 - 25) + 6(x - 5)
= x(x - 5)(x + 5) + 6(x - 5)
= (x - 5)(x2 + 5x + 6)
= (x - 5)(x2 + 2x + 3x + 6)
= (x - 5)[x(x + 2) + 3(x + 2)]
= (x - 5)(x + 2)(x + 3)
e) x3 - 7x - 6
= x3 - 9x + 2x - 6
= x(x2 - 9) + 2(x - 3)
= x(x - 3)(x + 3) + 2(x - 3)
= (x - 3)(x2 + 3x + 2)
= (x - 3)(x2 + x + 2x + 2)
= (x - 3)[x(x + 1) + 2(x + 1)]
= (x - 3)(x + 1)(x + 2)
f) x3 - 5x2 - 14x
= x3 + 2x2 - 7x2 - 14x
= x2(x + 2) - 7x(x + 2)
= (x + 2)(x2 - 7x)
= x(x + 2)(x - 7)
a) Ta có: \(a^4+a^2+1\)
\(=a^4+2a^2+1-a^2\)
\(=\left(a^2+1\right)^2-a^2\)
\(=\left(a^2-a+1\right)\left(a^2+a+1\right)\)
b) Ta có: \(a^4+a^2-2\)
\(=a^4-a^2+2a^2-2\)
\(=a^2\left(a^2-1\right)+2\left(a^2-1\right)\)
\(=\left(a^2-1\right)\left(a^2+2\right)\)
\(=\left(a-1\right)\left(a+1\right)\left(a^2+2\right)\)
c) Ta có: \(x^2+4x^2-5\)
\(=5x^2-5\)
\(=5\left(x^2-1\right)\)
\(=5\left(x-1\right)\left(x+1\right)\)
d) Ta có: \(x^3-19x-30\)
\(=x^3+2x^2-2x^2-4x-15x-30\)
\(=x^2\left(x+2\right)-2x\left(x+2\right)-15\left(x+2\right)\)
\(=\left(x+2\right)\left(x^2-2x-15\right)\)
\(=\left(x+2\right)\left(x^2-5x+3x-15\right)\)
\(=\left(x+2\right)\left[x\left(x-5\right)+3\left(x-5\right)\right]\)
\(=\left(x+2\right)\left(x-5\right)\left(x+3\right)\)
e) Ta có: \(x^3-7x-6\)
\(=x^3+2x^2-2x^2-4x-3x-6\)
\(=x^2\left(x+2\right)-2x\left(x+2\right)-3\left(x+2\right)\)
\(=\left(x+2\right)\left(x^2-2x-3\right)\)
\(=\left(x+2\right)\left(x^2-3x+x-3\right)\)
\(=\left(x+2\right)\left[x\left(x-3\right)+\left(x-3\right)\right]\)
\(=\left(x+2\right)\left(x-3\right)\left(x+1\right)\)
f) Ta có: \(x^3-5x^2-14x\)
\(=x\left(x^2-5x-14\right)\)
\(=x\left(x^2-7x+2x-14\right)\)
\(=x\left[x\left(x-7\right)+2\left(x-7\right)\right]\)
\(=x\left(x-7\right)\left(x+2\right)\)
