13: \(x^3-6x^2+12x-8\)

\(=x^3-3\cdot x^2\cdot2+3\cdot x\cdot2^2-2^3\)

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

14: \(8x^3+12x^2y+6xy^2+y^3\)

\(=\left(2x\right)^3+3\cdot\left(2x\right)^2\cdot y+3\cdot2x\cdot y^2+y^3\)

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

16: \(x^{10}-1=\left(x^5-1\right)\left(x^5+1\right)=\left(x-1\right)\left(x^4+x^3+x^2+x+1\right)\left(x+1\right)\left(x^4-x^3+x^2-x+1\right)\)

17: \(x^2-9=x^2-3^2=\left(x-3\right)\left(x+3\right)\)

18: \(4x^2-25=\left(2x\right)^2-5^2=\left(2x-5\right)\left(2x+5\right)\)

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

20: \(9x^2+6xy+y^2=\left(3x\right)^2+2\cdot3x\cdot y+y^2=\left(3x+y\right)^2\)

21: \(6x-9-x^2=-\left(x^2-6x+9\right)\)

\(=-\left(x^2-2\cdot x\cdot3+3^2\right)\)

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

22: \(x^2+4xy+4y^2=x^2+2\cdot x\cdot2y+\left(2y\right)^2=\left(x+2y\right)^2\)

23: \(\left(x+y\right)^2-\left(x-y\right)^2\)

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

\(=2x\cdot2y=4xy\)

24: \(\left(x+y+z\right)^2-4z^2\)

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

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

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

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

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

\(=2x\left(4x+2\right)=4x\left(2x+1\right)\)

26: \(x^3y^3+125=\left(xy\right)^3+5^3\)

\(=\left(xy+5\right)\left(x^2y^2-5xy+25\right)\)

27: \(8x^3-y^3-6xy\left(2x-y\right)\)

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

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

\(=\left(2x-y\right)\left(4x^2-4xy+y^2\right)=\left(2x-y\right)^3\)

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

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

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

a: Xét ΔABC có D,E lần lượt là trung điểm của AC,AB

=>DE là đường trung bình của ΔABC

=>DE//BC và \(DE=\dfrac{BC}{2}=2\left(cm\right)\)

Xét hình thang BEDC có

M,N lần lượt là trung điểm của EB,DC

=>MN là đường trung bình của hình thang BEDC

=>MN//ED//BC và \(MN=\dfrac{ED+BC}{2}=\dfrac{2+4}{2}=3\left(cm\right)\)

b: Xét ΔBED có MP//ED
nên \(\dfrac{MP}{ED}=\dfrac{BM}{BE}=\dfrac{1}{2}\)

=>\(MP=\dfrac{1}{2}ED=\dfrac{1}{2}\cdot\dfrac{1}{2}\cdot BC=\dfrac{1}{4}BC\)

Xét ΔCED có NQ//ED
nên \(\dfrac{NQ}{ED}=\dfrac{CN}{CD}=\dfrac{1}{2}\)

=>\(NQ=\dfrac{1}{2}ED=\dfrac{1}{2}\cdot\dfrac{1}{2}\cdot BC=\dfrac{1}{4}BC\)

\(MN=\dfrac{1}{2}\left(ED+BC\right)=\dfrac{1}{2}\left(\dfrac{1}{2}BC+BC\right)=\dfrac{1}{2}\cdot\dfrac{3}{2}BC=\dfrac{3}{4}BC\)

=>\(MP+PQ+QN=\dfrac{3}{4}BC\)

=>\(PQ=\dfrac{3}{4}BC-\dfrac{1}{4}BC-\dfrac{1}{4}BC=\dfrac{1}{4}BC\)

Do đó:MP=PQ=QN

\(a.4x-6y=2\left(2x-3y\right)\\ b.x^2+6x+9-y^2\\ =\left(x+3\right)^2-y^2\\ =\left(x-y+3\right)\left(x+y+3\right)\\ c.4x^2-9y^2\\ =\left(2x\right)^2-\left(3y\right)^2\\ =\left(2x-3y\right)\left(2x+3y\right)\\ d.x^2-x-y^2+y\\ =\left(x^2-y^2\right)-\left(x-y\right)\\ =\left(x+y\right)\left(x-y\right)-\left(x-y\right)\\ =\left(x-y\right)\left(x+y-1\right)\\ e.x^2-4y^2-6x+9\\ =\left(x^2-6x+9\right)-4y^2\\ =\left(x-3\right)^2-\left(2y\right)^2\\ =\left(x-2y-3\right)\left(x+2y-3\right)\\ f.x^4-y^4+4y^2-4\\ =x^4-\left(y^4-4y^2+4\right)\\ =\left(x^2\right)^2-\left(y^2-2\right)^2\\ =\left(x^2-y^2+2\right)\left(x^2+y^2-2\right)\)

Câu 2: 

\(A=x^2-10x+1\\ =\left(x^2-10x+25\right)-24\\ =\left(x-5\right)^2-24\ge-24\forall x\)

Dấu "=" xảy ra: `x-5=0<=>x=5` 

\(B=x^2-5x+2\\ =\left(x^2-5x+\dfrac{25}{4}\right)-\dfrac{17}{4}\\ =\left(x^2-2\cdot x\cdot\dfrac{5}{2}+\left(\dfrac{5}{2}\right)^2\right)-\dfrac{17}{4}\\ =\left(x-\dfrac{5}{2}\right)^2-\dfrac{17}{4}\ge-\dfrac{17}{4}\forall x\)

Dấu "=" xảy ra: `x-5/2=0<=>x=5/2` 

Câu 2

A = x² - 10x + 1

= x² + 2.x.5 + 25 - 24

= (x + 5)² - 24

Do (x + 5)² ≥ 0 với mọi x ∈ R

⇒ (x + 5)² - 24 ≥ -24 (với mọi x ∈ R)

Vậy GTNN của A là -24 khi x = -5

B = x² - 5x + 2

Bạn xem lại đề, là $13x^3$ hay $13x^2$ vậy?

Lời giải:
Trước hết ta cần nắm 1 số tính chất:

- Một scp lẻ khi chia 8 dư 1 (bạn có thể xét mô đun 4 của số đó để chứng minh)

- Một scp khi chia 5 dư $0,1$ hoặc $4$.

----------------------

Ta có: $2n+7$ là scp lẻ nên $2n+7\equiv 1\pmod 8$

$\Rightarrow 2n+6\equiv 0\pmod 8$

$\Rightarrow n+3\equiv 0\pmod 4$

$\Rightarrow n$ lẻ. 

$\Rightarrow 3n+10$ cũng là scp lẻ.

$\Rightarrow 3n+10\equiv 1\pmod 8$

$\Rightarrow 3n+9\equiv 0\pmod 8$

$\Rightarrow 3(n+3)\equiv 0\pmod 8\Rightarrow n+3\equiv 0\pmod 8(*)$

Lại có:

Đặt $2n+7=a^2, 3n+10=b^2$ với $a,b$ là số tự nhiên.

$\Rightarrow 2n+7+3n+10=a^2+b^2$

$\Rightarrow a^2+b^2=5n+17\equiv 2\pmod 5$

Ta thấy $a^2\equiv 0,1,4\pmod 5; b^2\equiv 0,1,4\pmod 5$

Do đó để $a^2+b^2\equiv 2\pmod 5$ thì chỉ khi $a^2, b^2\equiv 1\pmod 5$

$\Rightarrow 3n+10\equiv 1\pmod 5$

$\Rightarrow 3n+9\equiv 0\pmod 5$

$\Rightarrow 3(n+3)\equiv 0\pmod 5$

$\Rightarrow n+3\equiv 0\pmod 5(**)$

Từ $(*); (**)$ mà $(5,8)=1$ nên $n+3\vdots 40$.

Yesterday, I went to (1) the park with my friends. We had a picnic under (2) a big tree and played frisbee. In the afternoon, we went for (3) a walk around the lake. We saw (4) an interesting bird that we had never seen before. It had (5) a bright red beak and (6) a long tail. We also saw the ducks swimming in the lake. After our walk, we had (7) an ice cream cone and then went (8) chỗ này không cần điền home

21. finishes (lịch trình có sẵn -> THì HTĐ, chủ ngữ số ít)

22. to design (prefer + to V: Chỉ sở thích | prever + V(ing) + to V(ing): thích làm gì hơn làm gì)

23. hates (Chỉ sở thích -> HTĐ, chủ ngữ số ít)

24. doesn't like (Chỉ sở thích -> thì HTĐ, chủ ngữ số ít, thể phủ định đi với doesn't)

25. Does ... start (Lịch trình có sẵn -> Thì HTĐ, thể câu hỏi chủ ngữ số ít đi với does, động từ chính đứng sau chủ ngữ và nguyên thể)

21will finish 22 designing 23hates 24doesn't like 25Will /start

1. don't stay

2. don't wash

3. doesn't do

4. don't go

5. doesn't open

6. doesn't eat

7. don't chat

8. doesn't use

9. don't skate

10. doesn't throw

1. I don’t stay/ doesn’t stay at home.

2. We don’t wash/ doesn’t wash the family car.

3. Doris don’t do/ doesn’t do her homework.

4. They don’t go/ doesn’t go to bed at 8.30 p.m.

5. Kevin don’t open/ doesn’t open his workbook.

6. Our hamster don’t eat/ doesn’t eat apples.

7. You don’t chat/ doesn’t chat with your friends.

8. She don’t use/ doesn’t use a ruler.

9. Max, Frank and Steve don’t skate/ doesn’t skate in the yard.

10. The boy don’t throw/ doesn’t throw stones.