Giúp vs

a) \(\left(\right. x + y \left.\right)^{3} - \left(\right. x + y \left.\right) \left(\right. x^{2} - x y + y^{2} \left.\right) = 3 x y \left(\right. x + y \left.\right)\)

Giải:

Bắt đầu với vế trái của phương trình:

\(\left(\right. x + y \left.\right)^{3} - \left(\right. x + y \left.\right) \left(\right. x^{2} - x y + y^{2} \left.\right)\)

Bước 1: Mở rộng \(\left(\right. x + y \left.\right)^{3}\):

\(\left(\right. x + y \left.\right)^{3} = x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}\)

Bước 2: Mở rộng \(\left(\right. x + y \left.\right) \left(\right. x^{2} - x y + y^{2} \left.\right)\):

\(\left(\right. x + y \left.\right) \left(\right. x^{2} - x y + y^{2} \left.\right) = x \left(\right. x^{2} - x y + y^{2} \left.\right) + y \left(\right. x^{2} - x y + y^{2} \left.\right)\)\(= x^{3} - x^{2} y + x y^{2} + y x^{2} - x y^{2} + y^{3}\)\(= x^{3} + y^{3} + \left(\right. y x^{2} - x^{2} y \left.\right) = x^{3} + y^{3}\)

Bước 3: Trừ các biểu thức:

\(\left(\right. x + y \left.\right)^{3} - \left(\right. x + y \left.\right) \left(\right. x^{2} - x y + y^{2} \left.\right) = \left(\right. x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3} \left.\right) - \left(\right. x^{3} + y^{3} \left.\right)\)\(= 3 x^{2} y + 3 x y^{2}\)\(= 3 x y \left(\right. x + y \left.\right)\)

Vậy, phương trình đã đúng:

\(\left(\right. x + y \left.\right)^{3} - \left(\right. x + y \left.\right) \left(\right. x^{2} - x y + y^{2} \left.\right) = 3 x y \left(\right. x + y \left.\right)\)

b) \(B = \left(\right. 3 x + 2 \left.\right) \left(\right. 9 x^{2} - 6 x + 4 \left.\right) - 3 \left(\right. 9 x^{3} - 2 \left.\right)\)

Giải:

Bước 1: Mở rộng \(\left(\right. 3 x + 2 \left.\right) \left(\right. 9 x^{2} - 6 x + 4 \left.\right)\):

\(\left(\right. 3 x + 2 \left.\right) \left(\right. 9 x^{2} - 6 x + 4 \left.\right) = 3 x \left(\right. 9 x^{2} - 6 x + 4 \left.\right) + 2 \left(\right. 9 x^{2} - 6 x + 4 \left.\right)\)\(= 27 x^{3} - 18 x^{2} + 12 x + 18 x^{2} - 12 x + 8\)\(= 27 x^{3} + 8\)

Bước 2: Mở rộng \(3 \left(\right. 9 x^{3} - 2 \left.\right)\):

\(3 \left(\right. 9 x^{3} - 2 \left.\right) = 27 x^{3} - 6\)

Bước 3: Trừ hai biểu thức:

\(B = \left(\right. 27 x^{3} + 8 \left.\right) - \left(\right. 27 x^{3} - 6 \left.\right) = 8 + 6 = 14\)

Vậy, \(B = 14\).

c) \(C = \left(\right. x - 2 \left.\right) \left(\right. x^{2} - 2 x + 4 \left.\right) - \left(\right. x^{3} - 7 \left.\right)\)

Giải:

Bước 1: Mở rộng \(\left(\right. x - 2 \left.\right) \left(\right. x^{2} - 2 x + 4 \left.\right)\):

\(\left(\right. x - 2 \left.\right) \left(\right. x^{2} - 2 x + 4 \left.\right) = x \left(\right. x^{2} - 2 x + 4 \left.\right) - 2 \left(\right. x^{2} - 2 x + 4 \left.\right)\)\(= x^{3} - 2 x^{2} + 4 x - 2 x^{2} + 4 x - 8\)\(= x^{3} - 4 x^{2} + 8 x - 8\)

Bước 2: Trừ biểu thức \(x^{3} - 7\):

\(C = \left(\right. x^{3} - 4 x^{2} + 8 x - 8 \left.\right) - \left(\right. x^{3} - 7 \left.\right)\)\(C = x^{3} - 4 x^{2} + 8 x - 8 - x^{3} + 7\)\(C = - 4 x^{2} + 8 x - 1\)

Vậy, \(C = - 4 x^{2} + 8 x - 1\).

d) \(D = \left(\right. x + 1 \left.\right)^{3} - \left(\right. x - 1 \left.\right) \left(\right. x^{2} + x + 1 \left.\right) - 3 x \left(\right. x + 1 \left.\right)\)

Giải:

Bước 1: Mở rộng \(\left(\right. x + 1 \left.\right)^{3}\):

\(\left(\right. x + 1 \left.\right)^{3} = x^{3} + 3 x^{2} + 3 x + 1\)

Bước 2: Mở rộng \(\left(\right. x - 1 \left.\right) \left(\right. x^{2} + x + 1 \left.\right)\):

\(\left(\right. x - 1 \left.\right) \left(\right. x^{2} + x + 1 \left.\right) = x \left(\right. x^{2} + x + 1 \left.\right) - 1 \left(\right. x^{2} + x + 1 \left.\right)\)\(= x^{3} + x^{2} + x - x^{2} - x - 1\)\(= x^{3} - 1\)

Bước 3: Mở rộng \(3 x \left(\right. x + 1 \left.\right)\):

\(3 x \left(\right. x + 1 \left.\right) = 3 x^{2} + 3 x\)

Bước 4: Trừ các biểu thức:

\(D = \left(\right. x^{3} + 3 x^{2} + 3 x + 1 \left.\right) - \left(\right. x^{3} - 1 \left.\right) - \left(\right. 3 x^{2} + 3 x \left.\right)\)\(D = x^{3} + 3 x^{2} + 3 x + 1 - x^{3} + 1 - 3 x^{2} - 3 x\)\(D = 2\)

Vậy, \(D = 2\).

e) \(E = 3 \left(\right. x - 1 \left.\right) \left(\right. x^{2} + x + 1 \left.\right) + x \left(\right. x + 1 \left.\right) - x \left(\right. x^{2} + x + 1 \left.\right)\)

Giải:

Bước 1: Mở rộng \(3 \left(\right. x - 1 \left.\right) \left(\right. x^{2} + x + 1 \left.\right)\):

\(3 \left(\right. x - 1 \left.\right) \left(\right. x^{2} + x + 1 \left.\right) = 3 \left(\right. x \left(\right. x^{2} + x + 1 \left.\right) - \left(\right. x^{2} + x + 1 \left.\right) \left.\right)\)\(= 3 \left(\right. x^{3} + x^{2} + x - x^{2} - x - 1 \left.\right) = 3 \left(\right. x^{3} - 1 \left.\right)\)\(= 3 x^{3} - 3\)

Bước 2: Mở rộng \(x \left(\right. x + 1 \left.\right)\):

\(x \left(\right. x + 1 \left.\right) = x^{2} + x\)

Bước 3: Mở rộng \(x \left(\right. x^{2} + x + 1 \left.\right)\):

\(x \left(\right. x^{2} + x + 1 \left.\right) = x^{3} + x^{2} + x\)

Bước 4: Trừ các biểu thức:

\(E = \left(\right. 3 x^{3} - 3 \left.\right) + \left(\right. x^{2} + x \left.\right) - \left(\right. x^{3} + x^{2} + x \left.\right)\)\(E = 3 x^{3} - 3 + x^{2} + x - x^{3} - x^{2} - x\)\(E = 2 x^{3} - 3\)

Vậy, \(E = 2 x^{3} - 3\).

g) \(9 x \left(\right. x + 1 \left.\right)^{3} + \left(\right. x - 1 \left.\right)^{3} = 2 x^{3}\)

Giải:

Mở rộng biểu thức và kiểm tra tính đúng đắn:

\(9 x \left(\right. x + 1 \left.\right)^{3} = 9 x \left(\right. x^{3} + 3 x^{2} + 3 x + 1 \left.\right) = 9 x^{4} + 27 x^{3} + 27 x^{2} + 9 x\)\(\left(\right. x - 1 \left.\right)^{3} = x^{3} - 3 x^{2} + 3 x - 1\)

Cộng cả hai biểu thức:

\(9 x \left(\right. x + 1 \left.\right)^{3} + \left(\right. x - 1 \left.\right)^{3} = 9 x^{4} + 27 x^{3} + 27 x^{2} + 9 x + x^{3} - 3 x^{2} + 3 x - 1\)\(= 9 x^{4} + 28 x^{3} + 24 x^{2} + 12 x - 1\)

So với \(2 x^{3}\), ta thấy biểu thức không đúng. Có thể bài toán có lỗi. Nếu có sự nhầm lẫn, bạn có thể điều chỉnh lại nhé!

h) \(\left(\right. x + 3 \left.\right) \left(\right. x^{2} - 3 x + 9 \left.\right) = x \left(\right. x^{2} - 3 x + 9 \left.\right) = x \left(\right. x^{2} + 4 \left.\right) - 1\)

a) \(\left(3x-5\right)\left(2x+3\right)-\left(2x-3\right)\left(3x+7\right)-2x\left(x-4\right)\)

\(=\left(6x^2-x-15\right)-\left(6x^2+5x-21\right)-\left(2x^2-8x\right)\)

\(=6x^2-x-15-6x^2-5x+21-2x^2+8x\)

\(=-2x^2+2x+6\)

\(=-2\left(x^2-x-3\right)\)

b) \(\left(x^2+2\right)^2-\left(x+2\right)\left(x-2\right)\left(x^2+4\right)\)

\(=\left(x^2+2\right)^2-\left(x^2-4\right)\left(x^2+4\right)\)

\(=\left(x^2+2\right)^2-\left(x^4-16\right)\)

\(=\left(x^4+4x^2+4\right)-\left(x^4-16\right)\)

\(=x^4+4x^2+4-x^4+16\)

\(=4x^2+20\)

\(=4\left(x^2+5\right)\)

c) \(\left(2x-y\right)^2-2\left(x+3y\right)^2-\left(1+3x\right)\left(3x-1\right)\)

\(=\left(4x^2-4xy+y^2\right)-2\left(x^2+6xy+9y^2\right)-\left(9x^2-1\right)\)

\(=4x^2-4xy+y^2-2x^2-16xy-18y^2-9x^2+1\)

\(=-7x^2-20xy-17y^2+1\)

d) \(\left(x^2-1\right)^3-\left(x^4+x^2+1\right)\left(x^2-1\right)\)

\(=\left(x^6-3x^4+3x^2-1\right)-\left(x^6-1\right)\)

\(=x^6-3x^4+3x^2-1-x^6+1\)

\(=-3x^4+3x^2\)

\(=-3x^2\left(x^2-1\right)\)

\(=-3x^2\left(x-1\right)\left(x+1\right)\)

e) \(\left(2x-1\right)^2-2\left(4x^2-1\right)+\left(2x+1\right)^2\)

\(=\left(2x-1\right)^2-2\left(2x-1\right)\left(2x+1\right)+\left(2x+1\right)^2\)

\(=\left[\left(2x-1\right)-\left(2x+1\right)\right]^2\)

\(=\left(2x-1-2x-1\right)^2\)

\(=\left(-2\right)^2=4\)

g) \(\left(x-y+z\right)^2+\left(y-z\right)^2-2\left(x-y+z\right)\left(z-y\right)\)

\(=\left(x-y+z\right)^2+2\left(x-y+z\right)\left(y-z\right)+\left(y-z\right)^2\)

\(=\left(x-y+z+y+z\right)^2\)

\(=\left(x+2z\right)^2\)

h) \(\left(2x+3\right)^2+\left(2x+5\right)^2-\left(4x+6\right)\left(2x+5\right)\)

\(=\left(2x+3\right)^2-2\left(2x+3\right)\left(2x+5\right)+\left(2x+5\right)^2\)

\(=\left[\left(2x+3\right)-\left(2x+5\right)\right]^2\)

\(=\left(2x+3-2x-5\right)^2\)

\(=\left(-2\right)^2=4\)

i) \(5x^2-\dfrac{10x^3+15x^2-5x}{-5x}-3\left(x+1\right)\)

\(=5x^2-\dfrac{-5x\left(-2x^2-3x+1\right)}{-5x}-3\left(x+1\right)\)

\(=5x^2-\left(-2x^2-3x+1\right)-3\left(x+1\right)\)

\(=5x^2+2x^2+3x-1-3x-3\)

\(=7x^2-4\)

a, 3x - 3y = 3( x- y )

b, x² - x =x(x - 1)

c, 3(x - y) - 5x(y - x)

= 3(x - y) + 5x(x - y)

= ( x - y)(3 + 5x)

d, x(y - 1) - y(y - 1)

= (x - y)(y - 1)

e, 10x(x - y)-8y( y - x)

= 10x(x - y) + 8y(x - y)

= (10y + 8x)(x - y)

f, 2x² +5x³ +xy

= x(2x + 5x² + y)

g, 14x²y - 21xy² +28x²y²

= 7xy(2x - 3y + 4xy)

h, x² - 3x + 2

= x² - x - 2x + 2

= x(x - 1)- 2(x - 1)

= (x - 2)(x - 1)

i, x² - x - 6

x² + 2x - 3x - 6

x(x + 2) - 3(x + 2)

(x + 2)(x - 3)

k, x² + 5x+6

= x² - x + 6x + 6

=x(x - 1) + 6(x + 1)

= x(x - 1) - 6(x - 1)

= (x - 6)(x - 1)

l,x² - 4x + 3

= x² - x - 3x + 3

= x(x - 1) - 3(x - 1)

= (x - 3)(x - 1)

m, x² + 5x +4

= x² + x + 4x + 4

= x(x + 1) + 4(x + 1)

= (x + 4)(x + 1)

Hướng dẫn:

a, b, c, d, e, f, g: Phương pháp phân phối đưa thừa số chung ra ngoài

h, i, k, l, m : Tách hạng tử rồi nhóm

Bạn làm ra, đoạn nào không hiểu có thể inbox riêng để hoàn thành từng câu

a) \(A=\dfrac{\left(-2\right)^5}{\left(-2\right)^3}=\left(-2\right)^{5-3}=\left(-2\right)^2=4\)

b) \(y\ne0:B=\dfrac{\left(-y\right)^7}{\left(-y\right)^3}=\left(-y\right)^{7-3}=\left(-y\right)^4=y^4\)

c) \(x\ne0:C=\dfrac{\left(x\right)^{12}}{\left(-x\right)^{10}}=\left(x\right)^{12-10}=\left(x\right)^2=x^4\)

d) \(x\ne0:D=\dfrac{2x^6}{\left(2x\right)^3}=\dfrac{2x^6}{8x^3}=\dfrac{1}{4}\left(x\right)^{6-3}=\dfrac{1}{4}\left(x\right)^3\)

e) \(x\ne0:E=\dfrac{\left(-3x\right)^5}{\left(-3x\right)^2}=\left(-3x\right)^{5-2}=\left(-3x\right)^3=-27x^3\)

f) \(x,y\ne0:F=\dfrac{\left(xy^2\right)^4}{\left(xy^2\right)^2}=\left(xy^2\right)^{4-2}=\left(xy^2\right)^2=x^2y^4\)

i) \(x\ne-2:I=\dfrac{\left(x+2\right)^9}{\left(x+2\right)^6}=\left(x+2\right)^{9-6}=\left(x+2\right)^3\)

A),(-2)⁵:(-2)³=(-2)²=4

B) (-y)⁷ :(-y)³=y⁴

A = 2\(x^2\) + 3y\(^2\) - 8\(x\) - 6y + 15

A = 2(\(x^2\) - 4\(x\) + 4) + 3(y\(^2-2y+1\)) + 6

A = 2.(\(x-2)^2\) + 3(y - 1)\(^2\) + 4

Vì (\(x-2)^2\) ≥ 0; ∀ \(x\); (y -1)\(^2\) ≥ 0 ∀ y

⇒ 2.(\(x-2)^2\) ≥ 0 ∀ \(x\); 3(y - 1)\(^2\) + 4 ≥ y ∀ y

2.(\(x-2)^2\) + 3(y - 1)\(^2\) + 4 ≥ 4; Dấu bằng xảy ra khi:

\(\begin{cases}x-2=0\\ y-1=0\end{cases}\)

\(\begin{cases}x=2\\ y=1\end{cases}\)

Vậy A đạt giá trị nhỏ nhất là 4 khi (\(x;y\)) = (2; 1)

quy đồng nha bạn

Bài 1:

\(6x^2-2\left(x-y\right)^2-6y^2\)

\(=6\left(x-y\right)\left(x+1\right)-2\left(x-y\right)^2\)

\(=2\left(x-y\right)\left(3x+3-x+y\right)\)

\(=2\left(x-y\right)\left(2x+3+y\right)\)

Bài 2:

\(P=\left(3x-1\right)^2+2\left(3x-1\right)\left(x+1\right)+\left(x+1\right)^2\)

\(=\left(3x-1-x-1\right)^2\)

\(=\left(2x-2\right)^2\)(1)

b) Thay \(x=\frac{9}{4}\)vào (1) ta được: 

\(\left(2.\frac{9}{4}-2\right)^2\)

\(=\frac{25}{4}\)

Vậy giá trị của P \(=\frac{25}{4}\)khi \(x=\frac{9}{4}\)

Bài 3:

Ta có: \(M=x^2+4x+5\)

\(=\left(x+2\right)^2+1\)

Vì \(\left(x+2\right)^2\ge0;\forall x\)

\(\Rightarrow\left(x+2\right)^2+1\ge0+1;\forall x\)

Hay \(M\ge1;\forall x\)

Dấu"="xảy ra \(\Leftrightarrow\left(x+2\right)^2=0\)

                       \(\Leftrightarrow x=-2\)

Vậy \(M_{min}=1\Leftrightarrow x=-2\)

Bài 1 : trên là sai nha mình làm lại

\(6x^2-2\left(x-y\right)^2-6y^2\)

\(=6\left(x-y\right)\left(x+y\right)-2\left(x-y\right)^2\)

\(=2\left(x-y\right)\left(3x+3y-x+y\right)\)

\(=2\left(x-y\right)\left(2x+4y\right)\)

\(=4\left(x-y\right)\left(x+2y\right)\)

a)\(\frac{4x+1}{3x}\)+\(\frac{2x-3}{6x}\)=\(\frac{2\left(4x+1\right)}{2.3x}\)+\(\frac{2x-3}{6x}\)=\(\frac{8x+2}{6x}\)+\(\frac{2x-3}{6x}\)=\(\frac{8x+2+2x-3}{6x}\)=\(\frac{10x-1}{6x}\)

b)\(\frac{x^2-y^2}{6x^2y^2}\):\(\frac{x+y}{3xy}\)=\(\frac{\left(x+y\right)\left(x-y\right)}{6x^2y^2}\) . \(\frac{3xy}{x+y}\)=\(\frac{\left(x+y\right)\left(x-y\right)3xy}{6x^2y^2\left(x+y\right)}\)=\(\frac{x-y}{2xy}\)