phân tích các đa thức sau thành nhân tử
1/ Tách 1 hạng tử thành nhiều hạng tử
a)a^4+a^2+1
b)a^4+a^2-2
c)x^4+4x^2-5
d)x^3-19x-30
e)x^3-7x-6
f)x^2yz+5xyz-14yz
2/ Thêm bớt cùng một hạng tử
a)x^4+4
b)x^4+64
c)x^8+x^7+1
d)x^8+x^4+1
e)X^5+x+1
f)x^3+x^2+4
g)x^4+2x^2-24
h)x^3-2x-4
i)a^4+4ab^4
1/ Tách 1 hạng tử thành nhiều hạng tử
a)a^4+a²+1
= a^4 – a + a² + a + 1
= (a^4 – a) + (a² + a + 1)
= a(a – 1)(a² + a + 1) + (a² + a + 1)
= (a² + a + 1)(a² – a + 1)
b)a^4 + a² – 2
= a^4 – a³ + a³ – a² + 2a² – 2a + 2a – 2
= (a^4 – a³) + (a³ – a²) + (2a² – 2a) + (2a – 2)
= a³(a – 1) + a²(a – 1) + 2a(a – 1) + 2(a – 1)
= (a – 1)(a³ + a² + 2a + 2)
= (a – 1)[(a³ + a²) + (2a + 2)]
= (a – 1)[a²(a + 1) + 2(a +1)]
= (a – 1)(a + 1)(a² + 2)
c)x^4 + 4x² – 5
= x^4 – x³ + x³ – x² + 5x² – 5x + 5x – 5
= (x^4 – x³) + (x³ – x²) + (5x² – 5x) + (5x – 5)
= x³(x – 1) + x²( x – 1) + 5x( x – 1) + 5( x – 1)
= (x – 1)(x³ + x² + 5x + 5)
= (x-1)[x²(x+1) + 5(x+1)]
= (x-1)(x+1)(x²+5)
d)x³-19x-30
= x³ + 2x² – 2x² – 4x – 15x – 30
= x²(x+2) – 2x(x+2) – 15(x+2)
= (x+2)(x² – 2x -15)
= (x+2)(x² + 3x – 5x – 15)
= (x+2)[x(x+3) – 5(x+3)]
=(x+2)(x+3)(x-5)
e)x³-7x-6
= x³ – 3x² + 3x² – 9x + 2x – 6
= x²(x -3) + 3x(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)x²yz+5xyz-14yz
= yz(x² + 5x – 14)
= yz(x² – 2x + 7x – 14)
= yz[x(x-2) + 7(x-2)]
= yz(x-2)(x+7)
2/ Thêm bớt cùng một hạng tử
a)x^4+4
=x^4 + 4x² + 4 – 4x²
= (x² + 2)² – (2x)²
= (x² + 2 – 2x)(x² + 2 + 2x)
b)x^4+64
= x^4 + 16x² + 64 – 16x²
= (x² + 8)² – (4x)²
= (x² – 4x + 8)(x² + 4x + 8)
c)x^8+x^7+1
= x^8 + x^7 + x^6 + x^5 + x^4 + x³ + x² + x – x^6 – x^5 – x^4 – x³ – x² – x + 1
= x^6(x²+x+1) + x³(x²+x+1) + (x² + x + 1) – x^4(x²+x+1)-x(x²+1+1)
= (x²+x+1)(x^6-x^4+x³-x+1)
d)x^8+x^4+1
= x^8 – x^5 + x^5 – x² + x^4 – x + x² + x + 1
= (x^8 – x^5) + (x^5 – x²) + (x^4 – x) + (x² + x + 1)
= x^5(x-1)(x²+x+1) + x²(x-1)(x²+x+1) + (x² + x + 1)
= (x²+x+1)(x^6-x^5 + x³ – x² + 1)
e)x^5+x+1
= x^5 – x² + x² + x + 1
= x²(x-1)(x²+x+1) + (x²+x+1)
=(x²+x+1)(x³-x²+1)
f)x³+x²+4
= x³ + 2x² – x² – 2x + 2x + 4
= x²(x+2) – x(x+2) + 2(x+2)
= (x+2)(x²-x+2)
g)x^4+2x²-24
= x^4 – 2x³ + 2x³ – 4x² + 6x² – 12x + 12x – 24
= x³(x-2) + 2x²(x-2) + 6x(x-2) + 12(x-2)
= (x-2)(x³ + 2x² + 6x + 12)
= (x-2)[x²(x+2) + 6(x+2)]
= (x-2)(x+2)(x²+6)
h)x³-2x-4
= x³ – 2x² + 2x² – 4x + 2x – 4
= x²(x-2) + 2x(x-2) + 2(x-2)
= (x-2)(x² + 2x + 2)
i)a^4+4ab^4
= a(a³ + 4b^4)