phân tích da thức sau thành nhân tử (x+y)^5-x^5-y^5
áp dung nhị thức newton ta có :
\(\left(x+y\right)^5-x^5-y^5=5x^4y+10x^3y^2+10x^2y^3+5xy^4=5xy\left(x^3+2x^2y+2xy^2+y^3\right)\)
mà \(x^3+y^3=\left(x+y\right)\left(x^2-xy+y^2\right)\); \(2x^2y+2yx^2=2xy\left(x+y\right)\)
do đó \(A=5xy\left(x+y\right)\left(x^2+xy+y^2\right)\)
(x+y+z)^3-x^3-y^3-z^3
\(=\left\{\left[\left(x+y\right)+z\right]-\left(x+y\right)^3-z^3\right\}+3x^2y+3xy^2\)\(=\left[3\left(x+y\right)z\left(x+y+z\right)\right]+3xy\left(x+y\right)\)
\(=3\left(x+y\right)\left(xz+zy+z^2+xy\right)=3\left(x+y\right)\left(y+z\right)\left(z+x\right)\)
1.PTĐTTNT
a, x^2-2xy-25-y^2
b, x( x-1)+y (1-x)
c, 7x+7y-(x-y)
d, x^4+y^4
2, Chứng minh rằng:
a, x^2-5x+3≥0
b, -x^2+3x-4<0 với mọi x
Bài 1:
a) \(x^2-2xy-25+y^2\) (Sửa đề)
\(=x^2-2xy+y^2-25\)
\(=\left(x-y\right)^2-5^2\)
\(=\left(x-y-5\right)\left(x-y+5\right)\)
Vậy ...
b) \(x\left(x-1\right)+y\left(1-x\right)\)
\(=x\left(x-1\right)-y\left(x-1\right)\)
\(=\left(x-1\right)\left(x-y\right)\)
Vậy ...
c) \(7x+7y-\left(x+y\right)\) (Sửa đề)
\(=7\left(x+y\right)-\left(x+y\right)\)
\(=\left(x+y\right)\left(7-1\right)\)
\(=6\left(x+y\right)\)
Vậy ...
d) \(x^4+y^4\)
\(=\left(x^2\right)^2+\left(y^2\right)^2\)
\(=\left(x^2+y^2\right)^2-2x^2y^2\)
\(=\left(x^2+y^2-\sqrt{2}xy\right)\left(x^2+y^2+\sqrt{2}xy\right)\)
Vậy ...
Phân tích đa thức thành nhân tử ( phối hợp các phương pháp )
( xy + 4ab )2 + 4 ( ay - bx )2
Các bạn giải nhanh cho mình nha . Mình đag cần gấp .
\(\left(xy+4ab\right)^2+4\left(ay-bx\right)^2\)
\(=x^2y^2+2xy.4ab+4^2a^2b^2+4\left(a^2y^2-2aybx+b^2x^2\right)\)
\(=x^2y^2+8aybx+16a^2b^2+4a^2y^2-8aybx+4b^2x^2\)
\(=x^2y^2+16a^2b^2+4a^2y^2+4b^2x^2\)
\(=\left(x^2y^2+4b^2x^2\right)+\left(16a^2b^2+4a^2y^2\right)\)
\(=x^2\left(y^2+4b^2\right)+4a^2\left(4b^2+y^2\right)\)
\(=\left(y^2+4b^2\right)\left(x^2+4a^2\right)\)
\(3x^4-48\)
\(=\left(3x^4-6x^3\right)+\left(6x^3-12x^2\right)+\left(12x^2-24x\right)+\left(24x-48\right)\)
\(=3x^3\left(x-2\right)+6x^2\left(x-2\right)+12x\left(x-2\right)+24\left(x-2\right)\)
\(=\left(x-2\right)\left[\left(3x^3+6x^2\right)+\left(12x+24\right)\right]\)
\(=\left(x-2\right)\left[3x^2\left(x+2\right)+12\left(x+2\right)\right]\)
\(=\left(x-2\right)\left(x+2\right)\left(3x^2+12\right)\)
Phân tích đa thức thành nhân tử ( phối hợp các phương pháp )
x3 - 8x2 + 16x - 8
Các bạn giải gấp cho mình nha . Mình đag cần rất gấp .
1) Phân tích đa thức thành nhân tử:
a) \(\dfrac{1}{8}\)x3y3-27
b) \(\dfrac{8}{125}\)x3+\(\dfrac{1}{8}\)y3
c) 0.008x6-27y3
d) (2x+y)3-(x-y)3
2) Tính giá trị biểu thức
A= x3y2-x2y3-2x+2y
B= 5x2-3x+3y-5y2
Phân tích đa thức thành nhân tử:
a) 3x(x+1)2-5x2(x+1)+7(x+1)
b)(2x+1)2-(x-1)2
c)9(x+5)2-(x-7)2
Phân tích đa thức thành nhân tử
a) (x+1)(x+3)(x+5)(x+7)+15
b) \(x^3+4x^2-5x\)
c) \(x^3-5x^2+8x-4\)
\(x^3-8x^2+16x-8\)
\(=x^3-2x^2-6x^2+12x+4x-8\)
\(=x^2\left(x-2\right)-6x\left(x-2\right)+4\left(x-2\right)\)
\(=\left(x-2\right)\left(x^2-6x+4\right)\)
Câu a :
\(3x\left(x+1\right)^2-5x^2\left(x+1\right)+7\left(x+1\right)\)
\(=\left(x+1\right)\left(-2x^2+3x+7\right)\)
Câu b :
\(\left(2x+1\right)^2-\left(x-1\right)^2\)
\(=\left(2x+1-x+1\right)\left(2x+1+x-1\right)\)
\(=3x\left(x+2\right)\)
Câu c :
\(9\left(x+5\right)^2-\left(x-7\right)^2\)
\(=\left(3x+15\right)^2-\left(x-7\right)^2\)
\(=\left(3x+15-x+7\right)\left(3x+15+x-7\right)\)
\(=\left(2x+22\right)\left(4x+8\right)\)
\(=8\left(x+11\right)\left(x+2\right)\)
\(a.\left(x+1\right)\left(x+3\right)\left(x+5\right)\left(x+7\right)+15\)
\(=\left(x+1\right)\left(x+7\right).\left(x+3\right)\left(x+5\right)+15\)
\(=\left(x^2+8x+7\right)\left(x^2+8x+15\right)+15\)
\(=\left(x^2+8x+11-4\right)\left(x^2+8x+11+4\right)+15\)
\(=\left(x^2+8x+11\right)^2-4^2+15\)
\(=\left(x^2+8x+11\right)-1\)
\(=\left(x^2+8x+11-1\right)\left(x^2+8x+11+1\right)\)
\(=\left(x^2+8x+10\right)\left(x^2+8x+12\right)\)
\(=\left(x^2+8x+10\right)\left(x^2+2x+6x+12\right)\)
\(=\left(x^2+8x+10\right)\left[x\left(x+2\right)+6\left(x+2\right)\right]\)
\(=\left(x+2\right)\left(x+6\right)\left(x^2+8x+10\right)\)
\(b.x^3+4x^2-5x=x^3-x^2+5x^2-5x=x^2\left(x-1\right)+5x\left(x-1\right)=\left(x-1\right)\left(x^2+5x\right)=x\left(x-1\right)\left(x+5\right)\)
\(c.x^3-5x^2+8x-4=x^3-x^2-4x^2+4x+4x-4=x^2\left(x-1\right)-4x\left(x-1\right)+4\left(x-1\right)=\left(x-1\right)\left(x^2-4x+4\right)=\left(x-1\right)\left(x-2\right)^2\)
Bài 1:
a) \(\dfrac{1}{8}x^3y^3-27\)
\(=\left(\dfrac{1}{2}xy\right)^3-3^3\)
\(=\left(\dfrac{1}{2}xy-3\right)\left[\left(\dfrac{1}{2}xy\right)^2+\dfrac{1}{2}xy.3+3^2\right]\)
\(=\left(\dfrac{1}{2}xy-3\right)\left(\dfrac{1}{4}xy+\dfrac{3}{2}xy+9\right)\)
\(=\left(\dfrac{1}{2}xy-3\right)\left(\dfrac{7}{4}xy+9\right)\)
b) \(\dfrac{8}{125}x^3+\dfrac{1}{8}y^3\)
\(=\left(\dfrac{2}{5}x\right)^3+\left(\dfrac{1}{2}y\right)^3\)
\(=\left(\dfrac{2}{5}x+\dfrac{1}{2}y\right)\left[\left(\dfrac{2}{5}x\right)^2-\dfrac{2}{5}x.\dfrac{1}{2}y+\left(\dfrac{1}{2}y\right)^2\right]\)
\(=\left(\dfrac{2}{5}x+\dfrac{1}{2}y\right)\left(\dfrac{4}{25}x-\dfrac{1}{5}xy+\dfrac{1}{4}y\right)\)
c) \(0.008x^6-27y^3\)
\(=\left(\dfrac{1}{5}x^2\right)^3-\left(3y\right)^3\)
\(=\left(\dfrac{1}{5}x^2-3y\right)\left[\left(\dfrac{1}{5}x^2\right)^2+\dfrac{1}{5}x^2.3y+\left(3y\right)^2\right]\)
\(=\left(\dfrac{1}{5}x^2-3y\right)\left(\dfrac{1}{25}x^4+\dfrac{3}{5}x^2y+9y^2\right)\)
d) \(\left(2x+y\right)^3-\left(x-y\right)^3\)
\(=\left[\left(2x+y\right)-\left(x-y\right)\right]\left[\left(2x+y\right)^2+\left(2x+y\right)\left(x-y\right)+\left(x-y\right)^2\right]\)
\(=\left(2x+y-x+y\right)\left(4x^2+4xy+y^2+2x^3-2xy+xy-y^2+x^2-2xy+y^2\right)\)
\(=\left(x-2y\right)\left(4x^2+2x^3+xy\right)\)
Phân tích thành nhâu tử:
a) a(b2 + c2 + bc) + b(c2 + a2 +ac) + c(a2 +b2 + ab)
b) (a + b + c)(ab + bc + ca) - abc
c) a(a+2b)3 - b(2a+b)3
Phân tích đa thức thành nhân tử ( phối hợp các phương pháp )
2x3 + 16
ab( x - y )3 + 8ab
Các bạn giải gấp cho mình nha . Mình đag cần gấp .
\(2x^3+16=2\left(x^3+8\right)=2\left(x+2\right)\left(x^2-2x+4\right)\)
\(ab\left(x-y\right)^3+8ab=ab\left(\left(x-y\right)^3+8\right)=ab\left(x-y+2\right)\left(\left(x-y\right)^2-2\left(x-y\right)+4\right)=ab\left(x-y+2\right)\left(x^2-2xy+y^2-2x+2y+4\right)\)
a) \(a\left(b^2+c^2+bc\right)+b\left(c^2+a^2+ac\right)+c\left(a^2+b^2+ab\right)\)
\(=ab^2+ac^2+abc+bc^2+ba^2+abc+ca^2+cb^2+abc\)
\(=\left(ab^2+abc+ba^2\right)+\left(ac^2+ca^2+abc\right)+\left(bc^2+abc+cb^2\right)\)
\(=ab\left(b+c+a\right)+ac\left(c+a+b\right)+bc\left(c+a+b\right)\)
\(=\left(a+b+c\right)\left(ab+ac+bc\right)\)
b) \(\left(a+b+c\right)\left(ab+bc+ca\right)-abc\)
\(=ab^2+ac^2+abc+bc^2+ba^2+abc+ca^2+cb^2+abc-abc\)
\(=\left(ab^2+ba^2\right)+\left(ac^2+bc^2\right)+\left(abc+cb^2\right)+\left(abc+ca^2\right)\)
\(=ab\left(a+b\right)+c^2\left(a+b\right)+cb\left(a+b\right)+ca\left(b+a\right)\)
\(=\left(a+b\right)\left(ab+c^2+bc+ac\right)\)
\(=\left(a+b\right)\left[a\left(b+c\right)+c\left(c+b\right)\right]\)
\(=\left(a+b\right)\left(b+c\right)\left(a+c\right)\)
c) \(a\left(a+2b\right)^3-b\left(2a+b\right)^3\)
\(=a\left(a^3+3a^2.2b+3a4b^2+8b^3\right)-b\left(8a^3+3.4a^2.b+3.2a.b^2+b^3\right)\)
\(=a\left(a^3+6a^2b+12ab^2+8b^3\right)-b\left(8a^3+12a^2b+6ab^2+b^3\right)\)
\(=a^4+6a^3b+12a^2b^2+8b^3a-8a^3b-12a^2b^2-6ab^3-b^4\)
\(=a^4+6a^3b+8b^3a-8a^3b-6ab^3-b^4\)
\(=\left(a^4-b^4\right)+\left(6a^3b-6ab^3\right)+\left(8b^3a-8a^3b\right)\)
\(=\left(a^2-b^2\right)\left(a^2+b^2\right)+6ab\left(a^2-b^2\right)+8ab\left(b^2-a^2\right)\)
\(=\left(a^2-b^2\right)\left(a^2+b^2\right)+6ab\left(a^2-b^2\right)-8ab\left(a^2-b^2\right)\)
\(=\left(a^2-b^2\right)\left(a^2+b^2+6ab-8ab\right)\)
\(=\left(a-b\right)\left(a+b\right)\left(a^2+b^2-2ab\right)\)
\(=\left(a-b\right)\left(a+b\right)\left(a-b\right)^2\)
\(=\left(a-b\right)^3\left(a+b\right)\)
Phân tích đa thức thành nhân tử ( phối hợp các phương pháp )
9xy - 4a2xy
x2( a + b ) - 4ax - 4bx + 4a + 4b
Các bạn giải gấp cho mình nha . Mình đag cần gấp .
*\(9xy-4a^2xy=xy\left(9-4a^2\right)=xy\left(3-2a\right)\left(3+2a\right)\)
* \(x^2\left(a+b\right)-4ax-4bx+4a+4b\)
\(=x^2\left(a+b\right)-4x\left(a+b\right)+4\left(a+b\right)\)
\(=\left(a+b\right)\left(x^2-4x+4\right)\)
\(=\left(a+b\right)\left(x-2\right)^2\)
\(ab\left(a-b\right)+bc\left(b-c\right)+ac\left(c-a\right)\)
\(=ab\left(a-b\right)-bc\left[\left(a-b\right)+\left(c-a\right)\right]+ac\left(c-a\right)\)
\(=ab\left(a-b\right)-bc\left(a-b\right)-bc\left(c-a\right)+ac\left(c-a\right)\)
\(=\left(a-b\right)\left(ab-bc\right)+\left(c-a\right)\left(ac-bc\right)\)
\(=b\left(a-b\right)\left(a-c\right)+c\left(c-a\right)\left(a-b\right)\)
\(=b\left(a-b\right)\left(a-c\right)-c\left(a-c\right)\left(a-b\right)\)
\(=\left(a-b\right)\left(a-c\right)\left(b-c\right)\)
Phân tích đa thức thành nhân tử ( phối hợp các phương pháp )
16a3xy - 54b3xy4
Các bạn giải gấp cho mình nha . Mình đag cần rất gấp .
Giải:
\(16a^3xy-54b^3xy^4\)
\(=2xy\left(8a^3-27b^3y^3\right)\)
\(=2xy\left[\left(2a\right)^3-\left(3by\right)^3\right]\)
\(=2xy\left(2a-3by\right)\left[\left(2a\right)^2+\left(2a\right)\left(3by\right)+\left(3by\right)^2\right]\)
\(=2xy\left(2a-3by\right)\left(4a^2+6aby+9b^2y^2\right)\)
Vậy ...
Phân tích đa thức thành nhân tử ( phối hợp các phương pháp )
( a2 + 4 )2 - 16a2
Các bạn giải gấp cho mình nha . Mình đag cần gấp .
