[content] evaluation/easy_problems.jsonl (jsonl)
{"input": [{"role": "user", "content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.

Resolve the expression $\\frac{x^5 + x^4 - x^3 + 2x^2 + x}{(x-1)^2 (x^2 + 1)^2}$."}], "output": "<think>
Okay, so I need to solve this expression: (x^5 + x^4 - x^3 + 2x^2 + x) divided by (x-1)^2 (x^2 + 1)^2. The problem says to resolve it, which I think means to perform partial fraction decomposition. Alright, let's start by recalling how partial fractions work. 

First, the denominator is already factored into (x-1)^2 and (x^2 +1)^2. Since both of these are repeated irreducible factors, the partial fraction decomposition will have terms for each power up to the exponent. 

So for (x-1)^2, we'll have constants over (x-1) and (x-1)^2. Similarly, for (x^2 +1)^2, we'll have linear terms over (x^2 +1) and (x^2 +1)^2. So the general form should be something like:

(A)/(x-1) + (B)/(x-1)^2 + (Cx + D)/(x^2 +1) + (Ex + F)/(x^2 +1)^2

Now, the numerator of the original fraction is x^5 + x^4 - x^3 + 2x^2 + x. So when we express the original expression as the sum of these partial fractions, we can multiply both sides by the denominator (x-1)^2 (x^2 +1)^2 and equate coefficients.

Let me write that equation out:

x^5 + x^4 - x^3 + 2x^2 + x = A(x-1)(x^2 +1)^2 + B(x^2 +1)^2 + (Cx + D)(x-1)^2 (x^2 +1) + (Ex + F)(x-1)^2

Hmm, that seems a bit complicated, but let's expand each term step by step.

First, let's handle term by term.

Starting with A(x-1)(x^2 +1)^2:

First, expand (x^2 +1)^2: that's x^4 + 2x^2 +1. Then multiply by (x-1):

(x-1)(x^4 + 2x^2 +1) = x(x^4 + 2x^2 +1) -1(x^4 + 2x^2 +1)
= x^5 + 2x^3 + x - x^4 - 2x^2 -1
= x^5 - x^4 + 2x^3 -2x^2 + x -1

Multiply by A: A(x^5 -x^4 +2x^3 -2x^2 +x -1)

Next, term B(x^2 +1)^2:

(x^2 +1)^2 is x^4 +2x^2 +1, so B(x^4 +2x^2 +1) = Bx^4 + 2Bx^2 + B

Then, term (Cx + D)(x-1)^2 (x^2 +1):

First expand (x-1)^2: x^2 -2x +1

Multiply by (x^2 +1): (x^2 -2x +1)(x^2 +1) 

Multiply term by term:

x^2*(x^2 +1) = x^4 +x^2

-2x*(x^2 +1) = -2x^3 -2x

1*(x^2 +1) = x^2 +1

Combine those: x^4 +x^2 -2x^3 -2x +x^2 +1 = x^4 -2x^3 +2x^2 -2x +1

Now multiply by (Cx + D):

(Cx + D)(x^4 -2x^3 +2x^2 -2x +1) 

Let's distribute:

Cx*(x^4) = Cx^5

Cx*(-2x^3) = -2Cx^4

Cx*(2x^2) = 2Cx^3

Cx*(-2x) = -2Cx^2

Cx*(1) = Cx

D*(x^4) = Dx^4

D*(-2x^3) = -2Dx^3

D*(2x^2) = 2Dx^2

D*(-2x) = -2Dx

D*(1) = D

So combining all terms:

Cx^5 -2Cx^4 +2Cx^3 -2Cx^2 +Cx +Dx^4 -2Dx^3 +2Dx^2 -2Dx +D

Group like terms:

x^5: C

x^4: (-2C + D)

x^3: (2C - 2D)

x^2: (-2C + 2D)

x: (C - 2D)

constant: D

So this term becomes:

Cx^5 + (-2C + D)x^4 + (2C - 2D)x^3 + (-2C + 2D)x^2 + (C - 2D)x + D

Next, the term (Ex + F)(x-1)^2:

First, expand (x-1)^2 = x^2 - 2x +1

Multiply by (Ex + F):

Ex*(x^2 -2x +1) = Ex^3 -2Ex^2 +Ex

F*(x^2 -2x +1) = Fx^2 -2Fx +F

Combine:

Ex^3 -2Ex^2 +Ex + Fx^2 -2Fx +F

Group terms:

x^3: E

x^2: (-2E + F)

x: (E -2F)

constant: F

So this term is:

Ex^3 + (-2E + F)x^2 + (E -2F)x + F

Now, adding all the terms together (A's term, B's term, Cx+D term, Ex+F term):

Left side: x^5 +x^4 -x^3 +2x^2 +x

Right side:

From A: A x^5 -A x^4 +2A x^3 -2A x^2 +A x -A

From B: B x^4 +2B x^2 +B

From Cx+D: C x^5 + (-2C + D)x^4 + (2C - 2D)x^3 + (-2C + 2D)x^2 + (C - 2D)x + D

From Ex+F: E x^3 + (-2E + F)x^2 + (E -2F)x + F

Now, combine all these terms on the right side:

x^5 terms: A + C

x^4 terms: (-A + B -2C + D)

x^3 terms: (2A + 2C - 2D + E)

x^2 terms: (-2A + 2B -2C + 2D -2E + F)

x terms: (A + C -2D + E -2F)

constant terms: (-A + B + D + F)

Now, equate coefficients with the left side. The left side is:

x^5 +x^4 -x^3 +2x^2 +x +0

So setting up equations:

For x^5: A + C = 1

For x^4: (-A + B -2C + D) = 1

For x^3: (2A + 2C -2D + E) = -1

For x^2: (-2A + 2B -2C + 2D -2E + F) = 2

For x: (A + C -2D + E -2F) = 1

For constants: (-A + B + D + F) = 0

So we have a system of 6 equations:

1. A + C = 1

2. -A + B -2C + D = 1

3. 2A + 2C -2D + E = -1

4. -2A + 2B -2C + 2D -2E + F = 2

5. A + C -2D + E -2F = 1

6. -A + B + D + F = 0

Let's solve these step by step.

From equation 1: A + C =1, so C =1 - A.

Let's substitute C in other equations.

Equation 2: -A + B -2(1 - A) + D =1

Simplify: -A + B -2 + 2A + D =1

Which is (A) + B + D = 3 --> equation 2a: A + B + D =3

Equation 3: 2A + 2(1 - A) -2D + E = -1

Simplify: 2A +2 -2A -2D + E = -1 --> 0A +2 -2D + E = -1

Thus: -2D + E = -3 --> equation 3a: E = 2D -3

Equation 4: -2A + 2B -2(1 - A) +2D -2E + F =2

Simplify: -2A +2B -2 +2A +2D -2E + F =2

Simplify terms: (-2A +2A)=0, so 2B -2 +2D -2E + F =2

Which becomes 2B +2D -2E + F =4 --> equation 4a: 2B +2D -2E + F =4

Equation 5: A + (1 - A) -2D + E -2F =1

Simplify: 1 -2D + E -2F =1

Thus: -2D + E -2F =0 --> equation 5a: -2D + E -2F =0

Equation 6: -A + B + D + F =0 --> equation 6 remains as is.

Now, let's note from equation 3a: E =2D -3

Let's substitute E into equations 4a and 5a.

Equation 4a: 2B +2D -2(2D -3) + F =4

Simplify: 2B +2D -4D +6 + F =4 --> 2B -2D + F = -2 --> equation 4b: 2B -2D + F = -2

Equation 5a: -2D + (2D -3) -2F =0

Simplify: -2D +2D -3 -2F =0 --> -3 -2F=0 --> -2F=3 --> F= -3/2

So F= -3/2.

Now, substitute F into equation 6: -A + B + D + (-3/2)=0

So, -A + B + D = 3/2 --> equation 6a: -A + B + D = 3/2

From equation 2a: A + B + D =3

Now, subtract equation 6a from equation 2a:

(A + B + D) - (-A + B + D) =3 - 3/2

Left side: A + B + D + A - B - D = 2A

Right side: 3/2

Thus, 2A = 3/2 --> A= 3/4

Since A=3/4, then from equation1: C=1 - A=1 - 3/4=1/4

So C=1/4

Now, equation 3a: E=2D -3

We need to find D. Let's use equation 2a: A + B + D =3, with A=3/4.

So 3/4 + B + D =3 --> B + D= 3 - 3/4=9/4 --> equation 2b: B + D=9/4

From equation 6a: -A + B + D=3/2, with A=3/4:

-3/4 + B + D=3/2 --> B + D=3/2 +3/4= (6/4 +3/4)=9/4, which is consistent with equation 2b.

So, so far, we have:

A=3/4, C=1/4, F=-3/2

Now, equation 4b:2B -2D + F = -2

F=-3/2, so:

2B -2D -3/2 = -2

Multiply both sides by 2 to eliminate fractions:

4B -4D -3= -4

Thus,4B -4D= -1 --> divide by 4: B -D= -1/4 --> equation 4c: B= D -1/4

From equation 2b: B + D=9/4

Substitute B= D -1/4 into equation 2b:

(D -1/4) + D =9/4

2D -1/4=9/4

2D=10/4=5/2

D=5/4

Then, B=5/4 -1/4=4/4=1

So B=1, D=5/4

Now, E=2D -3=2*(5/4) -3=10/4 -12/4= -2/4= -1/2

So E= -1/2

Now, check equation5a: already used in substitution, but let's verify:

Equation5a: -2D + E -2F=0

-2*(5/4) + (-1/2) -2*(-3/2)= (-10/4 -1/2 +3)= (-5/2 -1/2 +3)= (-3 +3)=0. Correct.

Now, check equation3: already used.

Now, check equation4: Let me check equation4a: 2B +2D -2E + F=4?

But 2B=2*1=2, 2D=2*(5/4)=5/2, -2E= -2*(-1/2)=1, F= -3/2

So sum:2 +5/2 +1 -3/2=2 + (5/2 -3/2) +1=2 +1 +1=4. Correct.

Now, all equations are satisfied.

So now, we have all constants:

A=3/4

B=1

C=1/4

D=5/4

E= -1/2

F= -3/2

So the partial fractions decomposition is:

(A)/(x-1) + (B)/(x-1)^2 + (Cx + D)/(x^2 +1) + (Ex + F)/(x^2 +1)^2

Which is:

(3/4)/(x-1) + 1/(x-1)^2 + ( (1/4)x +5/4 )/(x^2 +1) + ( (-1/2)x -3/2 )/(x^2 +1)^2

We can write this with denominators as:

3/(4(x-1)) + 1/(x-1)^2 + (x +5)/(4(x^2 +1)) + (-x -3)/(2(x^2 +1)^2)

Alternatively, factoring out the constants:

= \\frac{3}{4(x-1)} + \\frac{1}{(x-1)^2} + \\frac{x + 5}{4(x^2 +1)} + \\frac{ -x -3 }{2(x^2 +1)^2}

To check, maybe we can substitute a value for x and see if both sides match.

Let's pick x=0. Compute original expression and the decomposition at x=0.

Original expression numerator at x=0:0 +0 -0 +0 +0=0. Denominator: (-1)^2*(0+1)^2=1*1=1. So expression is 0/1=0.

Decomposition at x=0:

3/(4*(-1)) + 1/((-1)^2) + (0 +5)/(4*(1)) + (0 -3)/(2*(1)^2)

= -3/4 +1 +5/4 -3/2

Convert to quarters:

-3/4 +4/4 +5/4 -6/4= ( -3 +4 +5 -6 )/4=0/4=0. Correct.

Another test, x=1. But x=1 makes denominator zero, so skip. Let's try x=2.

Original expression:

Numerator: 32 +16 -8 +8 +2= 50. Denominator: (1)^2*(4 +1)^2=1*25=25. So 50/25=2.

Decomposition at x=2:

3/(4*(1)) +1/(1)^2 + (2 +5)/(4*(5)) + (-2 -3)/(2*(25))

= 3/4 +1 +7/20 + (-5)/50

Convert to 100 denominators:

3/4=75/100, 1=100/100, 7/20=35/100, -5/50= -10/100.

Sum:75 +100 +35 -10=200. 200/100=2. Correct.

Another check x= -1.

Original expression numerator: -1 +1 +1 +2 +(-1)=2. Denominator: (-2)^2*(1 +1)^2=4*4=16. So 2/16=1/8.

Decomposition at x=-1:

3/(4*(-2)) +1/((-2)^2) + (-1 +5)/(4*(2)) + (1 -3)/(2*(2)^2)

= 3/(-8) +1/4 +4/8 + (-2)/8

= -3/8 +2/8 +4/8 -2/8= ( -3 +2 +4 -2 )/8=1/8. Correct.

All checks passed. So the partial fractions decomposition is correct.
</think>

The partial fraction decomposition of the given expression is:

\\boxed{\\frac{3}{4(x-1)} + \\frac{1}{(x-1)^2} + \\frac{x + 5}{4(x^2 + 1)} - \\frac{x + 3}{2(x^2 + 1)^2}}", "category": "math", "license": "cc-by-4.0", "reasoning": "on", "generator": "Qwen-2.5-32B-Instruct, DeepSeek-R1", "used_in_training": "Ultra, Super, Nano", "version": "v1.1", "system_prompt": "detailed thinking on"}