?

a: \(A\left(-1\right)=\left(-1\right)+\left(-1\right)^2+...+\left(-1\right)^{99}+\left(-1\right)^{100}\)

\(=\left(-1\right)+1+...+\left(-1\right)+1\)

=0+...+0

=0

=>x=-1 là nghiệm của A(x)

b: \(A\left(\dfrac{1}{2}\right)=\dfrac{1}{2}+\left(\dfrac{1}{2}\right)^2+\left(\dfrac{1}{2}\right)^3+...+\left(\dfrac{1}{2}\right)^{99}+\left(\dfrac{1}{2}\right)^{100}\)

=>\(2\cdot A\left(\dfrac{1}{2}\right)=1+\dfrac{1}{2}+...+\left(\dfrac{1}{2}\right)^{99}\)

=>\(2\cdot A\left(\dfrac{1}{2}\right)-A\left(\dfrac{1}{2}\right)=1+\dfrac{1}{2}+...+\left(\dfrac{1}{2}\right)^{99}-\dfrac{1}{2}-...-\left(\dfrac{1}{2}\right)^{100}\)

=>\(A\left(\dfrac{1}{2}\right)=1-\dfrac{1}{2^{100}}=\dfrac{2^{100}-1}{2^{100}}\)

Lời giải:

a. 

$A(x)=(x+x^2)+(x^3+x^4)+....+(x^{99}+x^{100})$

$=x(x+1)+x^3(x+1)+....+x^{99}(x+1)$

$=(x+1)(x+x^3+....+x^{99})$
Tại $x=-1$

$A(-1)=(-1+1)(x+x^3+...+x^{99})=0$
$\Rightarrow x=-1$ là nghiệm của $A(x)$

b.

Tại $x=\frac{1}{2}$

$A=\frac{3}{2}(\frac{1}{2}+\frac{1}{2^3}+....+\frac{1}{2^{99}})$

$\frac{1}{2^2}A=\frac{3}{2}(\frac{1}{2^3}+\frac{1}{2^5}+....+\frac{1}{2^{101}})$

$\Rightarrow A-\frac{1}{4}A=\frac{3}{2}(\frac{1}{2}-\frac{1}{2^{101}})$

$\Rightarrow \frac{3}{4}A=\frac{3}{2}(\frac{1}{2}-\frac{1}{2^{101}})$

$\Rightarrow A=2(\frac{1}{2}-\frac{1}{2^{101}})$

Lời giải:

$A(0)=a.0^2+b.0+c=7$

$\Rightarrow c=7$

$A(-1)=a(-1)^2+b(-1)+c=a-b+c=a-b+7=13$

$\Rightarrow a-b=6$

$\Rightarrow a=b+6$

$A(2)=a.2^2+b.2+c=4a+2b+c=4(b+6)+2b+7=6b+31=1$

$\Rightarrow b=-5$

$a=b+6=-5+6=1$

Vậy $A(x)=x^2-5x+7$

$A(x)=x^2-5x+7=(x^2-5x+2,5^2+0,75=(x-2,5)^2+0,75\geq 0+0,75>0$ với mọi $x$

$\Rightarrow A(x)\neq 0$ với mọi $x$

$\Rightarrow A(x)$ vô nghiệm.

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

\(=\dfrac{a}{x^5}+\dfrac{b}{x^3}+\dfrac{b}{x^2}+a\)

\(=\dfrac{a+bx^2+bx^3+ax^5}{x^5}\)

\(=\dfrac{f\left(x\right)}{x^5}\)

\(\Rightarrow f\left(\dfrac{1}{2021}\right)=\dfrac{f\left(2021\right)}{2021^5}=\dfrac{2021}{2021^5}=\dfrac{1}{2021^4}\)

It's hard to feed a large family.

a

Diện tích mảnh đất là \(\left(3x+2\right)\left(2x+4\right)\left(m^2\right)\)

Diện tích phần đất trồng hoa là \(x\left(x+1\right)\left(m^2\right)\)

Diện tích phần đất còn lại là:

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

\(=6x^2+12x+4x+8-x^2-x\)

\(=5x^2+15x+8\left(m^2\right)\)

1 haven't played - has been 

2 Have you chosen - bought

3 has forgot - didn't say

4 Have you sold - saw - bought

5 have lost - ran

6 has stopped

7 has changed

8 have forgot

9 arrested

10 has improved

1. The distance / my school/ is/ about 50 km/ between/ and Lao Cai/.

->.........The distance between my school and Lao Cai is about 50 km....................................................................................................................

2. You should / study  /harder/ taking English  test/ before/.

->.You should study harder before taking English test............................................................................................................................

3. /you/ visit New Zealand/ next month/ Will/ ?

->...Will you visit New Zealand next month?..........................................................................................................................

4. It/ my school/ is/ about 50 km/ from/ to Lao Cai/.

->.........It is about 50km from my school to Lao Cai....................................................................................................................

5. You should / speaking English/ practice /the UK/ going to//before/.

->...............You should practice speaking English before going to the UK..............................................................................................................

6. /you/ visit New Zealand/ next month/ Will/?

->..Will you visit New Zealand next month?...........................................................................................................................

7. should / use / to travel. /We / public transports //

………We should use public transports to travel…………………………………………………………………………………

8. for fun./ attends/ Everybody/ the festival //

……Everybody attends the festival for fun……………………………………………………………………………………