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DN
1
2 tháng 8 2017
\(x^3-x^2-21x+45\)
\(=\left(x^3-3x^2\right)+\left(2x^2-6x\right)+\left(-15x+45\right)\)
\(=x^2\left(x-3\right)+2x\left(x-3\right)-15\left(x-3\right)\)
\(=\left(x^2+2x-15\right)\left(x-3\right)\)
\(=\left[\left(x^2-3x\right)+\left(5x-15\right)\right]\left(x-3\right)\)
\(=\left[x\left(x-3\right)+5\left(x-3\right)\right]\left(x-3\right)\)
\(=\left(x+5\right)\left(x-3\right)^2\)
23 tháng 10 2021
Tham khảo:https://hoc247.net/hoi-dap/toan-8/phan-tich-da-thuc-x-7-x-2-1-thanh-nhan-tu-faq417522.html
23 tháng 10 2021
\(=x^7+x^6-x^6+x^5-x^5+x^4-x^4+x^3-x^3+x^2+x^2-x^2+x-x+1\\ =\left(x^7+x^6+x^5\right)-\left(x^6+x^5+x^4\right)+\left(x^4+x^3+x^2\right)-\left(x^3+x^2+x\right)+\left(x^2+x+1\right)\\ =\left(x^2+x+1\right)\left(x^5-x^4+x^2-x+1\right)\)
G
10 tháng 8 2023
\(4a^2b^2-\left(a^2+b^2-c^2\right)^2\)
\(=4a^2b^2-2ab\left(a^2+b^2-c^2\right)+2ab\left(a^2+b^2-c^2\right)-\left(a^2+b^2-c^2\right)^2\)
\(=2ab\left[2ab-\left(a^2+b^2-c^2\right)\right]+\left(a^2+b^2-c^2\right)\left[2ab-\left(a^2+b^2-c^2\right)\right]\)
\(=\left(2ab+a^2+b^2-c^2\right)\left(2ab-a^2-b^2+c^2\right)\)
\(=\left(a^2+ab+ab+b^2-c^2\right)\left[c^2-\left(a^2-ab-ab+b^2\right)\right]\)
\(=\left[a\left(a+b\right)+b\left(a+b\right)-c^2\right]\left[c^2-\left(a\left(a-b\right)-b\left(a-b\right)\right)\right]\)
\(=\left[\left(a+b\right)^2-c^2\right]\left[c^2-\left(a-b\right)^2\right]\)
\(=\left[\left(a+b\right)^2-c\left(a+b\right)+c\left(a+b\right)-c^2\right]\left[c^2+c\left(a-b\right)-c\left(a-b\right)-\left(a-b\right)^2\right]\)
\(=\left[\left(a+b\right)\left(a+b-c\right)+c\left(a+b-c\right)\right]\left[c\left(c+a-b\right)-\left(a-b\right)\left(c+a-b\right)\right]\)
\(=\left(a+b+c\right)\left(a+b-c\right)\left(c+a-b\right)\left(c-a+b\right)\)
28 tháng 12 2022
x^3-2x-4
=x^3-2x-8+4 (Ta thấy - 8 + 4 là bằng -4 nên ta thêm vào thì cũng giống nhau)
=(x^3-8)-(2x-4) (Nhóm hạng tử)
=(x-2)(x^2+2x+4)-2(x-2) \([\)(Hằng đẳng thức 6) và ta thấy -2 là nhân tử chung\(]\)
=(x-2)(x^2+2x+4-x+2) (Rút gọn)
=(x-2)(x^2+x+6)
17 tháng 11 2021
Đặt \(x^2+x+1=t\)
\(\left(x^2+x+1\right)\left(x^2+x+2\right)-12=t\left(t+1\right)-12=t^2+t-12=\left(t^2+t+\dfrac{1}{4}\right)-\dfrac{49}{4}=\left(t+\dfrac{1}{2}\right)^2-\left(\dfrac{7}{2}\right)^2=\left(t+\dfrac{1}{2}-\dfrac{7}{2}\right)\left(t+\dfrac{1}{2}+\dfrac{7}{2}\right)=\left(t-3\right)\left(t+4\right)=\left(x^2+x-2\right)\left(x^2+x+5\right)\)
17 tháng 11 2021
\(\left(x^2+x+1\right)\left(x^2+x+2\right)-12\)
= \(\left(x^2+x+1\right)\left[\left(x^2+x+1\right)+1\right]-12\)
= \(\left(x^2+x+1\right)^2\left(x^2+x+1\right)-12\)
= \(\left(x^2+x+1\right)\left(x^2+x+1\right)-3\left(x^2+x+1\right)+4\left(x^2+x+1\right)-4.3\)
= \(\left(x^2+x+1\right)\left(x^2+x-2\right)+4\left(x^2+x-2\right)\)
= \(\left(x^2+x+5\right)\left(x^2+x-2\right)\)
6 tháng 7 2017
Ta có \(x^4+4=\left(x^2\right)^2+2^2=\left(x^2+2\right)^2-2.x^2.2=\left(x^2+2\right)^2-\left(2x\right)^2\)
\(=\left(x^2+2x+2\right)\left(x^2-2x+2\right)\)
23 tháng 11 2021
\(ax-2x-a^2+2a=x\left(a-2\right)-a\left(a-2\right)=\left(a-2\right)\left(x-a\right)\)
32: \(16\left(x-y\right)^2-25\left(x+y\right)^2\)
\(=\left(4x-4y\right)^2-\left(5x+5y\right)^2\)
=(4x-4y-5x-5y)(4x-4y+5x+5y)
=(-x-9y)(9x+y)
31: \(4\left(x+6\right)^2-9\left(x+1\right)^2\)
\(=\left(2x+12\right)^2-\left(3x+3\right)^2\)
=(2x+12-3x-3)(2x+12+3x+3)
=(-x+9)(5x+15)
=5(-x+9)(x+3)
30: \(9\left(x-3\right)^2-25\)
\(=\left(3x-9\right)^2-25\)
=(3x-9-5)(3x-9+5)
=(3x-14)(3x-4)
29: \(\left(3x+1\right)^2-\left(x-2\right)^2\)
=(3x+1-x+2)(3x+1+x-2)
=(2x+3)(4x-1)
28: \(\left(x+9\right)^2-\left(3x+5\right)^2\)
=(x+9+3x+5)(x+9-3x-5)
=(-2x+4)(4x+14)
\(=-2\left(x-2\right)\cdot2\cdot\left(2x+7\right)=-4\left(x-2\right)\left(2x+7\right)\)
27: \(\left(2x-1\right)^2-\left(x-1\right)^2\)
=(2x-1-x+1)(2x-1+x-1)
=x(3x-2)
26: \(81-\left(4a+5\right)^2\)
\(=9^2-\left(4a+5\right)^2\)
=(9-4a-5)(9+4a+5)
=(-4a+4)(4a+14)
\(=-4\left(a-1\right)\cdot2\cdot\left(2a+7\right)=-8\cdot\left(a-1\right)\left(2a+7\right)\)
25: \(100-\left(2x-y\right)^2\)
\(=10^2-\left(2x-y\right)^2\)
=(10-2x+y)(10+2x-y)
24: \(\left(x+y\right)^2-x^2\)
=(x+y-x)(x+y+x)
=y(2x+y)
23: \(\left(3x-1\right)^2-4\)
\(=\left(3x-1\right)^2-2^2\)
=(3x-1-2)(3x-1+2)
=(3x-3)(3x+1)
=3(x-1)(3x+1)
64: \(5a^3b-10a^2b^2+5ab^3\)
\(=5ab\cdot a^2-5ab\cdot2ab+5ab\cdot b^2\)
\(=5ab\left(a^2-2ab+b^2\right)=5a\left(a-b\right)^2\)
63: \(a^2b+2ab^2+b^3\)
\(=b\left(a^2+2ab+b^2\right)\)
\(=b\left(a+b\right)^2\)
62: \(x^3-x^2y-a^2x+a^2y\)
\(=x^2\left(x-y\right)-a^2\left(x-y\right)\)
\(=\left(x-y\right)\left(x^2-a^2\right)=\left(x-y\right)\left(x-a\right)\left(x+a\right)\)
61: \(x^3-1+x-x^2\)
\(=x^3-x^2+x-1\)
\(=x^2\cdot\left(x-1\right)+\left(x-1\right)=\left(x-1\right)\left(x^2+1\right)\)
60: \(8x^3+ay-4x^2y-2ax\)
\(=8x^3-4x^2y+ay-2ax\)
\(=4x^2\left(2x-y\right)-a\left(2x-y\right)\)
\(=\left(2x-y\right)\left(4x^2-a\right)\)
58: 5ax-15ay-x+3y
=5a(x-3y)-(x-3y)
=(x-3y)(5a-1)
57: \(4xy-ay+8x^2-2ax\)
=y(4x-a)+2x(4x-a)
=(4x-a)(2x+y)
56: \(ax-2x-a^2+2a\)
\(=x\left(a-2\right)-a\left(a-2\right)\)
=(a-2)(x-a)