Phân tích đa thức thành nhân tử
a, x5+x+1
b, (x2+x)2-2(x2+x)-15
c, x4+5x3+10x-4
d, x8+x7+1
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12 tháng 7 2023
\(a,=\left(5x^3+10x\right)+\left(x^4-4\right)\\ =5x\left(x^2+2\right)+\left(x^2+2\right)\left(x^2-2\right)\\ =\left(x^2+2\right)\left(x^2+5x-2\right)\\ b,=\left(x+y\right)^3-3xy\left(x+y\right)+z^3-3xyz\\ =\left[\left(x+y\right)^3+z^3\right]-3xy\left(x+y+z\right)\\ =\left(x+y+z\right)\left[\left(x+y\right)^2-z\left(x+y\right)+z^2\right]-3xy\left(x+y+z\right)\\ =\left(x+y+z\right)\left(x^2+2xy+y-xz-yz+z^2-3xy\right)\\ =\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-zx\right)\)
\(c,=\left(x^8+x^7+x^6\right)-\left(x^7+x^6+x^5\right)+\left(x^5+x^4+x^3\right)-\left(x^4+x^3+x^2\right)+\left(x^2+x+1\right)\\ =\left(x^2+x+1\right)\left(x^6-x^5+x^3-x^2+1\right)\\ d,=\left(x^7+x^6+x^5\right)-\left(x^6+x^5+x^4\right)+\left(x^4+x^3+x^2\right)-\left(x^3+x^2+x\right)+\left(x^2+x+1\right)\\ =\left(x^2+x+1\right)\left(x^5-x^4+x^2-x+1\right)\\ e,=\left(x^{10}+x^9+x^8\right)-\left(x^9+x^8+x^7\right)+\left(x^7+x^6+x^5\right)-\left(x^6+x^5+x^4\right)+\left(x^5+x^4+x^3\right)-\left(x^3+x^2+x\right)+\left(x^2+x+1\right)\\ =\left(x^2+x+1\right)\left(x^{10}-x^7+x^5-x^4+x^3-x+1\right)\)
12 tháng 7 2023
a: =x^4+2x^2+5x^3+10x-2x^2-4
=(x^2+2)(x^2+5x-2)
b; =(x+y)^3+z^3-3xy(x+y)-3xyz
=(x+y+z)*(x^2+2xy+y^2-xz-yz+z^2)-3xy(x+y+z)
=(x+y+z)(x^2+y^2+z^2-xy-yz-xz)
c: =x^8+x^7+x^6-x^7-x^6-x^5+x^5+x^4+x^3-x^4-x^3-x^2+x^2+x+1
=(x^2+x+1)(x^6-x^5+x^3-x^2+1)
DN
4
1 tháng 9 2021
\(x^2\left(x+4\right)^2-\left(x+4\right)^2-\left(x^2-1\right)\\ =\left(x+4\right)^2\left(x^2-1\right)-\left(x^2-1\right)\\ =\left(x^2-1\right)\left[\left(x+4\right)^2-1\right]\\ =\left(x-1\right)\left(x+1\right)\left(x+4-1\right)\left(x+4+1\right)\\ =\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+5\right)\)
26 tháng 8 2021
a: \(x^4+4=\left(x^2-2x+2\right)\left(x^2+2x+2\right)\)
b: \(x^8+x^7+1\)
\(=x^8+x^7+x^6-x^6-x^5-x^4+x^5+x^4+x^3-x^3-x^2-x+x^2+x+1\)
\(=\left(x^2+x+1\right)\left(x^6-x^4+x^3-x+1\right)\)
c: \(x^8+x^4+1\)
\(=\left(x^8+2x^4+1\right)-x^4\)
\(=\left(x^4-x^2+1\right)\cdot\left(x^4+x^2+1\right)\)
\(=\left(x^4-x^2+1\right)\left(x^2+1-x\right)\left(x^2+1+x\right)\)
CM
1 tháng 1 2017
f(x) = x5 + 3x2 − 5x3 − x7 + x3 + 2x2 + x5 − 4x2 + x7
= (x5 + x5) + (3x2 + 2x2 – 4x2) + (-5x3 + x3) + (-x7 + x7)
= 2x5 + x2 – 4x3.
= 2x5 - 4x3 + x2
Đa thức có bậc là 5
g(x) = x4 + 4x3 – 5x8 – x7 + x3 + x2 – 2x7 + x4 – 4x2 – x8
= (x4 + x4) + (4x3 + x3) – (5x8 + x8) – (x7 + 2x7) + (x2 – 4x2)
= 2x4 + 5x3 – 6x8 – 3x7 – 3x2
= -6x8 - 3x7 + 2x4 + 5x3 - 3x2.
Đa thức có bậc là 8.
DN
2
4 tháng 9 2021
\(\left(x^2+6x-1\right)^2+2x^2+x^4+2\left(x^2+6x-1\right)\left(x^2+1\right)\)
\(\left(x^2+6x-1\right)^2+2\left(x^2+6x-1\right)\left(x^2+1\right)+\left(x^2+1\right)^2-1=\left(x^2+6x-1+x^2+1\right)^2-1=\left(2x^2+6x\right)^2-1=\left(2x^2+6x-1\right)\left(2x^2+6x+1\right)\)
4 tháng 9 2021
\(\left(x^2+6x-1\right)^2+2\left(x^2+6x-1\right)\left(x^2+1\right)+x^4+2x^2\)
\(=\left(x^2+6x-1\right)\left(x^2+6x-1+2x^2+2\right)+x^4+2x^2\)
\(=\left(x^2+6x-1\right)\left(3x^2+6x+1\right)+x^4+2x^2\)
\(=\left(2x^2+6x-1\right)\left(2x^2+6x+1\right)\)
DN
3
4 tháng 9 2021
\(\left(x^2+x\right)^2+4x^2+4x-12=\left[\left(x^2+x\right)^2+4\left(x^2+x\right)+4\right]-16=\left(x^2+x+2\right)-4^2=\left(x^2+x+2-4\right)\left(x^2+x+2+4\right)=\left(x^2+x-2\right)\left(x^2+x+6\right)=\left(x-1\right)\left(x+2\right)\left(x^2+x+6\right)\)
LN
4 tháng 9 2021
\(\left(x^2+x\right)^2+4x^2+4x-12\\ =\left(x^2+x+2\right)-4\\ =\left(x^2+x-2\right)\left(x^2+x+6\right)\)
DN
2
KK
4 tháng 9 2021
(1 + x2)2 - 4x(1 - x2)
= (1 + x2)(1 + x2) - 4x(1 - x2)
= (1 + x2 - 4x)(1 + x2 - 1 + x2)
= 2x2(x2 - 4x + 1)
4 tháng 9 2021
Ta có: \(\left(x^2+1\right)^2+4x\left(x^2-1\right)\)
\(=x^4+2x^2+1+4x^3-4x\)
\(=x^4+2x^3+2x^3+4x^2-2x^2-4x+1\)
\(=\left(x+2\right)\left(x^3+2x^2-2x\right)+1\)
DN
1
31 tháng 8 2021
\(x^2+3y^2-4xy+10x-12y+9\)
\(=\left(x^2-xy+x\right)+9x-3xy+3y^2-12y+9\)
\(=\left(x^2-xy+x\right)+\left(9x-9y+9\right)-3xy+3y^2-3y\)
\(=\left(x^2-xy+x\right)+\left(9x-9y+9\right)-\left(3xy-3y^2+3y\right)\)
\(=x\left(x-y+1\right)+9\left(x-y+1\right)-3y\left(x-y+1\right)\\ =\left(x-y+1\right)\left(x+9-3y\right)\)
4 tháng 9 2021
\(x^2-x-2020.2021=x^2+2020x-2021x-2020.2021=x\left(x+2020\right)-2021\left(x+2020\right)=\left(x+2020\right)\left(x-2021\right)\)
4 tháng 9 2021
\(x^2-x-2020\cdot2021\)
\(=\left(x-2021\right)\left(x+2020\right)\)
4 tháng 9 2021
\(x^5-x^4-30x^3=x^3\left(x^2-x-30\right)=x^3\left(x-6\right)\left(x+5\right)\)
a) x5 + x + 1
= x5 - x4 + x4 - x3 + x3 - x2 + x2 + x + 1
= ( x5 + x4 + x3) - ( x4 + x3 + x2) + (x2 + x +1)
= x3( x2 + x + 1) - x2( x2 + x + 1) + (x2 + x +1)
= ( x2 + x +1).( x3 - x2 + 1)
b) ( x2 + x)2 -2(x2 + x) -15
=( x2 + x)2 -2(x2 + x).1 + 1- 16
=( x2 + x - 1)2 - 42
=( x2 + x - 1 - 4).( x2 + x - 1+ 4)
=(x2 + x - 5).( x2 + x + 3)
c) x4 + 5x3 + 10x - 4
= (x2)2 - 22 + 5x.( x2 + 2)
=( x2 -2).(x2 + 2) + 5x.( x2 + 2)
= ( x2 + 2).(x2 -2 + 5x)
d) x8 + x7 + 1
= x8 + x7 + x6 - x6 + 1
= x6 ( x2 + x + 1) - ( x6 - 1)
= x6( x2 + x + 1) - ( x3 - 1).(x3 + 1)
= x6( x2 + x + 1) - ( x- 1).( x2 + x + 1).(x3 + 1)
= ( x2 + x + 1).[ x6 -( x- 1).(x3 + 1)]
= ( x2 + x + 1).( x6 - x4 + x3 - x +1)
cau b)
\(B=\left(x^2+x\right)^2-2\left(x^2+x\right)+1-16=\left(x^2+x-1\right)^2-4^2\)\(B=\left(x^2+x-3\right)\left(x^2+x+1\right)\)
\(B=\left[\left(x+\dfrac{1}{2}\right)^2-\left(\sqrt{\dfrac{13}{4}}\right)^2\right]\left(x^2+x+1\right)\)
\(B=\left(x+\dfrac{1-\sqrt{13}}{2}\right)\left(x+\dfrac{1+\sqrt{13}}{2}\right)\left(x^2+x+1\right)\)