Chứng tỏ rằng \(x^8-y^8⋮\left(x-y\right)\) và \(x^8-y^8⋮\left(x+y\right)\)
Chứng tỏ rằng \(x^8-y^8⋮\left(x-y\right)\) và \(x^8-y^8⋮\left(x+y\right)\)
thêm x;y thuộc z nhé
\(x^8-y^8=\left(x^4\right)^2-\left(y^4\right)^2=\left(x^4-y^4\right)\left(x^4+y^4\right)=\left(x^2-y^2\right)\left(x^2+y^2\right)\left(x^4+y^4\right)\)
\(=\left(x-y\right)\left(x+y\right)\left(x^2+y^2\right)\left(x^4+y^4\right)\)
vì \(x-y⋮x-y;x,y\in Z\Rightarrow\left(x-y\right)\left(x+y\right)\left(x^2+y^2\right)\left(x^4+y^4\right)⋮x-y\Rightarrow x^8-y^8⋮x-y\)
\(x+y⋮x+y;x,y\in Z\Rightarrow\left(x-y\right)\left(x+y\right)\left(x^2+y^2\right)\left(x^4+y^4\right)⋮x+y\Rightarrow x^8-y^8⋮x+y\)
Cho x > 1, y > 1. Chứng minh rằng : \(\dfrac{x^3+y^3-\left(x^2+y^2\right)}{\left(x-1\right)\left(y-1\right)}\) ≥ 8
Đặt \(\left(x-1;y-1\right)=\left(a;b\right)\Rightarrow\left(x;y\right)=\left(a+1;b+1\right)\)
\(VT=\dfrac{\left(a+1\right)^3+\left(b+1\right)^3-\left(a+1\right)^2-\left(b+1\right)^2}{ab}=\dfrac{a^3+a+b^3+b+2\left(a^2+b^2\right)}{ab}\)
\(VT\ge\dfrac{2a^2+2b^2+2\left(a^2+b^2\right)}{ab}=\dfrac{4\left(a^2+b^2\right)}{ab}\ge\dfrac{8ab}{ab}=8\)
Cho x=y+1. Chứng minh rằng:
a)\(x^3-y^3-3xy=1\)
b)\(\left(x+y\right)\left(x^2+y^2\right)\left(x^4+y^4\right)\left(x^8+y^8\right)=x^{16}-y^{16}\)
a. Do \(x=y-1\Rightarrow x-y=1\)
Ta có:
\(A=x^3-y^3-3xy=\left(x-y\right)^3+3xy\left(x-y\right)-3xy=1^3+3xy.1-3xy=1\left(đpcm\right)\)
b. \(B=\left(x-y\right)\left(x+y\right)\left(x^2+y^2\right)\left(x^4+y^4\right)\left(x^8+y^8\right)\)
(Do \(x-y=1\))
(Bạn áp dụng hằng đẳng thức \(x^2-y^2=\left(x-y\right)\left(x+y\right)\)vào bài toán)
Kết quả, \(B=x^{16}-y^{16}\left(đpcm\right)\)
a)\(x=y+1\Rightarrow x-y=1\Rightarrow\left(x-y\right)^3=1\)
Hay x3- 3xy(x-y) - y3=1 => x3- y3 -3xy =1
b) 1.(x+y)(x2+y2)(x4+y4)(x8+y8) = (x-y)(x+y)......................=(x2-y2)(x2+y2)..........=(x4-y4)(x4+y4)......=(x8-y8)(x8+y8) =x16-y16
Cho x, y, z thỏa mãn \(\dfrac{x}{2013}=\dfrac{y}{2014}=\dfrac{z}{2015}\). Chứng minh rằng: \(\left(x-z\right)^3=8\cdot\left(x-y\right)^2\left(y-z\right)\)
Áp dụng tc dtsbn:
\(\dfrac{x}{2013}=\dfrac{y}{2014}=\dfrac{z}{2015}=\dfrac{x-z}{-2}=\dfrac{y-z}{-1}=\dfrac{x-y}{-1}\\ \Leftrightarrow\dfrac{x-z}{2}=\dfrac{y-z}{1}=\dfrac{x-y}{1}\\ \Leftrightarrow x-z=2\left(y-z\right)=2\left(x-y\right)\\ \Leftrightarrow\left(x-z\right)^3=8\left(x-y\right)^3=8\left(x-y\right)^2\left(x-y\right)=8\left(x-y\right)^2\left(y-z\right)\)
Cho x, y, z >0. CMR:
a) \(2\left(x^8+y^8\right)\ge\left(x^3+y^3\right)\left(x^5+y^5\right)\)
b) \(3\left(x^8+y^8+z^8\right)\ge\left(x^3+y^3+z^3\right)\left(x^5+y^5+z^5\right)\)
a, Ta có: \(2\left(x^8+y^8\right)\ge\left(x^3+y^3\right)\left(x^5+y^5\right)\)
\(\Leftrightarrow x^8+y^8\ge x^5y^3+x^3y^5\)
Ta CM: \(\Leftrightarrow x^8+y^8\ge x^5y^3+x^3y^5\)
Áp dụng bđt Cô si:
\(x^8+x^8+x^8+x^8+x^8+y^8+y^8+y^8\ge8x^5y^3\) (*)
Tương tự, \(5y^3+3x^3\ge8x^3y^5\) (**)
Từ (*), (**) \(\Rightarrowđpcm\)
1.Cho x+y=7 và x.y=12. Tính giá trị của A=\(x^4+y^4\).
2.Cho ba số a,b,c khác 0 và a+b+c=0. Tính A=\(\dfrac{a^2}{bc}+\dfrac{b^2}{ac}+\dfrac{c^2}{ba}\)
3.Cho x=y+1. Chứng tỏ rằng \(\left(x+y\right)\left(x^2+y^2\right)\left(x^4+y^4\right)=x^8-y^8\)
Bài 3:
x=y+1 nên x-y=1
\(\left(x+y\right)\left(x^2+y^2\right)\left(x^4+y^4\right)\)
\(=\left(x+y\right)\cdot\left(x-y\right)\left(x^2+y^2\right)\left(x^4+y^4\right)\)
\(=\left(x^2-y^2\right)\left(x^2+y^2\right)\left(x^4+y^4\right)\)
=x^8-y^8
Rút gọn biểu thức với x - y = 1
\(\left(x+y\right)\left(x^2+y^2\right)\left(x^4+y^4\right)\left(x^8+y^8\right)\left(x^{16}+y^{16}\right)\)
Ta có \(x-y=1\)
\(=>x+y=\left(x+y\right).\left(x-y\right)\)
\(A=\left(x+y\right).\left(x-y\right).\left(x^2+y^2\right).\left(x^4+y^4\right)\)
\(A=\left(x^2-y^2\right).\left(x^2+y^2\right).\left(x^4+y^4\right)\)
\(A=\left(x^4-y^4\right).\left(x^4+y^4\right)\)
\(A=x^8-y^8\)
= \(-\left[\left(x-y\right)\left(x^2-y^2\right)\left(x^4-y^4\right)\left(x^8-y^8\right)\left(x^{16}-y^{16}\right)\right]\)
= \(-\left[\left(x-y\right)\left(x-y\right)^2\left(x-y\right)^4\left(x-y\right)^8\left(x-y\right)^{16}\right]\)
= \(-\left(1\cdot1^2\cdot1^4\cdot1^8\cdot1^{16}\right)\)
= -1
Cho x,y là các số thực lớn hơn 1. Chứng minh rằng:
\(\frac{\left(x^3+y^3\right)-\left(x^2+y^2\right)}{\left(x-1\right)\left(y-1\right)}\) lớn hơn hoặc bằng 8
1. Giải hệ phương trình \(\left\{{}\begin{matrix}3x^2+y^2+4xy=8\\\left(x+y\right)\left(x^2+xy+2\right)=8\end{matrix}\right.\)
2. chứng minh rằng với moi số nguyên n ta luôn có \(\left[\left(27n+5\right)^7+10\right]^7+\left[\left(10n+27\right)^7+5\right]^7+\left[\left(5n+10\right)^7+27\right]^7⋮42\)
1. \(\left\{{}\begin{matrix}3x^2+y^2+4xy=8\left(1\right)\\\left(x+y\right)\left(x^2+xy+2\right)=8\end{matrix}\right.\)
=> \(3x^2+3xy+xy+y^2=\left(x+y\right)\left(x^2+xy+2\right)\)
<=> \(\left(x+y\right)\left(3x+y\right)=\left(x+y\right)\left(x^2+xy+2\right)=0\)
<=> \(\left(x+y\right)\left(x^2+xy+2-3x-y\right)=0\)
<=> \(\left[{}\begin{matrix}x=-y\\x^2+xy+2-3x-y=0\end{matrix}\right.\)
TH1: x = -y thay vào pt (1), ta được:
3y2 + y2 - 4y2 = 8
<=> 0y = 8 (vô lí)
TH2: \(x^2+xy+2-3x-y=0\)
<=> x (x + y) - (x + y) - 2(x - 1) = 0
<=> (x - 1)(x + y) - 2(X - 1) = 0
<=> (x - 1)(x + y - 2) = 0
<=> \(\left[{}\begin{matrix}x=1\\x+y-2=0\end{matrix}\right.\)
Với x = 1 thay vào pt (1) -> 3 + y2 + 4y = 8
<=> y2 + 4y - 5 = 0 <=> (y + 5)(y - 1) = 0
<=> \(\left[{}\begin{matrix}y=-5\\y=1\end{matrix}\right.\)
Với x + y - 2 = 0 => x = 2 - y thay vào pt (1)
=> 3(2 - y)2 + y2 + 4(2 - y)y = 8
<=> 3y2 - 12y + 12 + y2 + 8 - 4y2 = 8
<=> 12 = 12y <=> y= 1 => x = 2 - 1 = 1
Vậy ....