X = 1 / (1 + √ 2), find √ (x ^ 3 + 2x ^ 2-x + 8)

X = 1 / (1 + √ 2), find √ (x ^ 3 + 2x ^ 2-x + 8)

x=1/(1+√2),
=√2-1
√(x^3+2x^2-x+8)
=√x(x+1)²-2x+8
=√2(√2-1)-2(√2-1)+8
=√8
=2√2

A problem in the first grade of junior high school, calculation: x ^ 2 + 2x + 4 - (x ^ 3) / (X-2)?
[(a^-1)-(b^-1)]/[(a^-2)+(a^-1)(b^-1)+(b^-2)]

x^2+2x+4-（x^3）/(x-2)
=(x-2)(x²+2x+4)/(x-2)-x³/(x-2)
=(x³-8)/(x-2)-x³/(x-2)
=-8/(x-2)
=8/(2-x)
The numerator denominator of (a ^ - 1) - (b ^ - 1) / (a ^ - 2) + (a ^ - 1) (b ^ - 1) + (b ^ - 2) is simultaneously multiplied by a & # 178; B & # 178;
=(ab²-a²b)/(b²+ab+a²)

6(1-x)-5(x-2)=2(2x+3) y-y-1/2=2-y+2/5

6(1-x)-5(x-2)=2(2x+3)
6-6x-5x+10=4x+6
-6x-5x-4x=6-6-10
-15x=-10
x=-10/(-15)
x=2/3
y-y-1/2=2-y+2/5
-1/2=2-y+2/5
y=2+2/5+1/2
y=29/10

The coefficients of x ^ 3 in the expansion of 1 - (x-1) + (x-1) ^ 2 - (x-1) ^ 3 +. + (x-1) ^ 20
Thank you~
The answer is - 5985

As an equal ratio sequence with 1 as the first term and (1-x) as the common ratio!
Use the sum formula!
[1 - (1-x) ^ 21] / X
The coefficient of x ^ 3 is
four
C 21 =[(-1)^17]*(21*20*19*18)/(1*2*3*4)
=-5985

The expansion of (x-1 / x) ^ 5 contains x ^ 3 term. The coefficient of the expansion is
Binomial coefficient of

(x-1/x）^5
X ^ 3 coefficient
C(5,1)x4(-1/x)=-C(5,2)x³=-5x³
Coefficient - 5
The binomial coefficient is C (5,1) = 5

What is the coefficient of X in the expansion of (1 + 2 √ x) ^ 3 (1 - & sup3; √ x) ^ 5?
Sup that place when the root of three times under X, I do not know why can not display
Can we use the binomial theorem in the conventional way.

（1+2 √x ）3（1-3√ x ）5=（1+6√ x +12x+8x√ x ）（1-3 √x ）5
So in the expansion of (1 + 2 √ x) 3 (1-3 √ x) 5, the term containing x is 1 × C53 (3 √ x) 3 + 12x = - 10x + 12xc50 = 2x,
So the coefficient of X is 2
So the option is C

Find the coefficient of (1 + x-x ^ 2-x ^ 3) ^ 5 with x ^ 3 term

=1+x-x2(1+x)
=(1+x)2（1-x）
（1+x-x^2-x^3)^5=(1+x)10(1-x)5
The coefficients with x ^ 3 term in the expansion are the constant term in (1 + x) 10 multiplied by the x ^ 3 term in (1-x) 5, and the X of (1 + x) 10 versus the X2 term in (1-x) 5. There are four groups
1*(-C5,3)+C10,1*C5,2+C10,2*(-C5,1)+C10,3=205

High school mathematics related point method and parameter method
Give a few typical questions, have train of thought to go

Correlation point method (moving point transfer method) for some more complex problems of exploring trajectory equation, we can first determine the trajectory equation of a point which is easy to obtain, and then take this point as the active point, and take the point on the obtained trajectory as the correlation point to obtain the trajectory equation
For example, it is known that P (4,0) is a point in the circle x2 + y2 = 36, a and B are two moving points on the circle, and satisfy ∠ APB = 90 ° to find the trajectory equation of vertex Q of rectangular apbq
In the parametric method, if x and Y in the coordinates (x, y) of the moving point change with the change of another variable, we can use this variable as a parameter to establish the parametric equation of the trajectory
For example, let points a and B be two moving points beyond the origin on the parabola y2 = 4PX (P > 0), and OA ⊥ ob, OM ⊥ AB are known. Find the trajectory equation of point m, and explain what curve it represents

What is the coefficient of the expansion x2 of (1-x) ^ 4 × (1 + x) ^ 4?

-4

The function f (x) = x & # 178; - 2lnx, G (x) = X-2 √ x.1 is known. It is proved that f (x) = g (x) + 2 has a unique solution when x > 0
2. When B ＞ - 1, if f (x) ≥ 2bx - (1 / X & # 178;), when 0 ＜ x ≤ 1, the constant holds, the value range of B is obtained

2>b≥1