# 3x+12×3+3x23x+12×3+3×2 Write the partial fraction decomposition of the rational expression.

Juliet dies by her own hand after awaking from the sleeping potion Friar Laurence gave her. When Juliet wakes up in her family mausoleum (her family, believing her dead, had placed her body there),…

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**THE SCARLET IBIS**

**What happens to scarlet ibis in the bleeding tree?**

Doodle spots the ibis first and when Brother goes out to investigate, he notes that the bird is perched “precariously” (unbalanced) and that its wings were hanging loosely. The ibis is on the…

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**THE PEARL**

**What quote in Chapter One or Two show what the doctor says about Kino and people like him in The…**

The doctor does not care about Kino because he is poor, and he refuses to treat the sick baby. Kino is suspicious of the doctor. Kino and his wife are aboriginals (Indians) and the doctor is one…

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**HISTORY**

**How did Jim Crow laws contribute to social unrest in the United States and what role did the…**

The term Jim Crow refers to the systematic segregation of the black and white races, primarily in the South. Segregation was deemed legal by the Supreme Court in the 1896 Plessy vs. Ferguson Case….

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**REFERENCE**

**How can a person become an educator on the eNotes website?**

Teachers can earn money on the eNotes web page in a couple of different ways. The first step is to apply to become an eNotes educator through the web page here. You will be asked to answer three…

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**MATH**

**How would you solve (z-2)(z+1)=0 using the Zero Product Property?**

The Zero Product Property is a property of real numbers that says that for any two real numbers x and y, if x*y = 0 then x = 0 or y = 0 (or both).Thus if we have (z-2)(z+1) = 0, then we know that…

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**SOCIAL SCIENCES**

**What are two aspects of U.S Imperialism that Twain disagrees with?**

Twain most clearly and famously criticizes imperialism in his 1901 essay “To the Person Sitting in Darkness”. He criticizes the imperialistic activities of several countries, including Germany and…

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**MATH**

**Write the equation in slope- intercept form for a line Slope -2/3; through (4,2)**

Hello! An equation of a straight line in a slope-intercept form is y=mx+b, where m is a slope. It is given that the line goes through the point (4, 2), i.e. y=2, x=4. Therefore 2=4m+b. m is also…

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**LAW AND POLITICS**

**Why are tampons and other feminine products taxed as a luxury item in many American states?…**

To first clarify the idea of taxing tampons, it is important to note that the ‘United States’ does not tax feminine products. That is because the federal government does not impose a sales tax. The…

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**MACBETH**

**What figure of speech is used by the captain when he says “disdaining fortune” in Act 1, Scene 1…**

Macbeth was not worried about what would happen. A figure of speech is no-literal language, or figurative language. It can be a simile, a metaphor, or an idiom. If it is an idiom, it would mean…

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**HISTORY**

**What are some positive and negative words about the American industrial Revolution?**

The American Industrial Revolution had some positive and negative characteristics associated with it. Some of the words that would reflect the positive characteristics would be economic growth,…

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**THE BOY IN THE STRIPED PAJAMAS**

**What are several geographic details that help establish the setting in The Boy in the Striped…**

The setting in a novel always has to do with two specific elements: setting of time and setting of place. The Boy in the Striped Pajamas is no exception, as it is set in a very specific place and…

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**LITERATURE**

**What is the summary for Mink River by Brian Doyle?**

As Doyle’s first fictional novel, we are introduced here to his very poetic (and often Gaelic) style as we learn about a little town on the coast of Oregon called Neawanaka containing the memorable…

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**LITERATURE**

**Why have writers like Tennyson continued to tell stories about King Arthur’s life and death?**

The legends of King Arthur have proven to be hugely influential to the British canon. The tales even endure into today, and have inspired countless iterations and reinterpretations of the classic…

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**OLIVER TWIST**

**How does Dickens present the character Oliver Twist as an innocent character in Oliver Twist?**

Oliver Twist is innocent because he is incorruptible. The subtitle of this book is “The Parish Boy’s Progress” for a reason. A progress is a type of story where the main character is not…

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**MACBETH**

**Who is more evil in Macbeth, Lady Macbeth or Macbeth? (P.E.E paragrapgh)**

Who is more evil, Macbeth or Lady Macbeth? This is certainly a matter of opinion, and Shakespeare gives readers the opportunity to decide for themselves who is “worse” or who is more responsible…

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**ADAM SMITH**

**According to Adam Smith, what are the benefits of specialization and the division of labor?**

Specialization of labor is very important to Adam Smith–so important, in fact, that it is the first subject that he addresses in detail in Wealth of Nations. He claims in the first chapter of the…

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**THE MIDWIFE’S APPRENTICE**

**How does the setting help set the mood/tone of the The Midwife’s Apprentice?**

The Midwife’s Apprentice is set in medieval England, and the realities of life in a small English village during that period of time contribute to the mood and tone of the story in many ways. Even…

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**EX-BASKETBALL PLAYER**

**Who was the speaker of “Ex-Basketball Player”?**

There is no way to know for sure who the speaker of “Ex-Basketball Player” is. The narrator never explains who he or she is. All that the reader knows for sure is that the speaker of the poem was…

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**THE MIRACLE WORKER**

**In The Miracle Worker, what are things Helen does that demonstrate her intelligence and…**

Helen cannot see or hear, but this does not make her unaware or unintelligent. One day, her mother puts on her bonnet because she is going out. Helen feels the bonnet on her mother’s head and…

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**ARTS**

**How typical is Toy Story of its genre?**

Pixar Studio’s Toy Story was revolutionary upon the film’s release in 1995 in terms of the studio’s use of computer generated imagery. The film was a landmark and a number of other studios and…

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**MATH**

**x3−x+3×2+x−2×3-x+3×2+x-2 Write the partial fraction decomposition of the rational…**

x3−x+3×2+x−2×3-x+3×2+x-2 Divide by applying long division method, x3−x+3×2+x−2=(x−1)+2x+1×2+x−2×3-x+3×2+x-2=(x-1)+2x+1×2+x-2 Now continue with the partial fraction of the remainder expression,…

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**MATH**

**2×3−4×2−15x+5×2−2x−82×3-4×2-15x+5×2-2x-8 Write the partial fraction decomposition of the…**

You need to decompose the fraction in simple irreducible fractions, such that: 2×3−4×2−15x+5×2−2x−8=2×3−4×2−15x+5(x−4)(x+2)2×3-4×2-15x+5×2-2x-8=2×3-4×2-15x+5(x-4)(x+2)

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**MATH**

**x3(x+2)2(x−2)2×3(x+2)2(x-2)2 Write the partial fraction decomposition of the rational expression.**

x3(x+2)2(x−2)2×3(x+2)2(x-2)2 Let x3(x+2)2(x−2)2=Ax+2+B(x+2)2+Cx−2+D(x−2)2×3(x+2)2(x-2)2=Ax+2+B(x+2)2+Cx-2+D(x-2)2 x3(x+2)2(x−2)2=A(x+2)(x−2)2+B(x−2)2+C(x−2)(x+2)2+D(x+2)2(x+2)2(x−2)2×3(x+2)2(x-2)2=A(x+2)(x-2)2+B(x-2)2+C(x-2)(x+2)2+D(x+2)2(x+2)2(x-2)2…

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**MATH**

**x2+x+2(x2+2)2×2+x+2(x2+2)2 Write the partial fraction decomposition of the rational…**

x2+x+2(x2+2)2×2+x+2(x2+2)2 Letx2+x+2(x2+2)2=Ax+Bx2+2+Cx+D(x2+2)2×2+x+2(x2+2)2=Ax+Bx2+2+Cx+D(x2+2)2 x2+x+2(x2+2)2=(Ax+B)(x2+2)+Cx+D(x2+2)2×2+x+2(x2+2)2=(Ax+B)(x2+2)+Cx+D(x2+2)2 ∴(x2+x+2)=(Ax+B)(x2+2)+Cx+D∴(x2+x+2)=(Ax+B)(x2+2)+Cx+D…

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**MATH**

**3x+12×3+3x23x+12×3+3×2 Write the partial fraction decomposition of the rational expression.**

3x+12×3+3x23x+12×3+3×2 To decompose this to partial fractions, factor the denominator. 2×3+3×2=x2(2x+3)2×3+3×2=x2(2x+3) Then, write a fraction for each factor. For the repeated factor x, form a partial…

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**MATH**

**4×2−12x(x+1)24×2-12x(x+1)2 Write the partial fraction decomposition of the rational expression.**

4×2−12x(x+1)24×2-12x(x+1)2 4×2−12x(x+1)2=A2x+Bx+1+C(x+1)24×2-12x(x+1)2=A2x+Bx+1+C(x+1)2 4×2−12x(x+1)2=A(x+1)2+B(2x)(x+1)+C(2x)2x(x+1)24×2-12x(x+1)2=A(x+1)2+B(2x)(x+1)+C(2x)2x(x+1)2 ∴(4×2−1)=A(x+1)2+B(2x)(x+1)+C(2x)∴(4×2-1)=A(x+1)2+B(2x)(x+1)+C(2x)…

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**MATH**

**3×2−7x−2×3−x3x2-7x-2×3-x Write the partial fraction decomposition of the rational…**

3×2−7x−2×3−x=3×2−7x−2x(x+1)(x−1)3×2-7x-2×3-x=3×2-7x-2x(x+1)(x-1) 3×2−7x−2×3−x=Ax+Bx+1+Cx−13×2-7x-2×3-x=Ax+Bx+1+Cx-1 Multiply through by the LDC x3−x.x3-x. 3×2−7x−2=A(x2−1)+Bx(x−1)+Cx(x+1)3×2-7x-2=A(x2-1)+Bx(x-1)+Cx(x+1)…

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**MATH**

**5−x2x2+x−15-x2x2+x-1 Write the partial fraction decomposition of the rational expression.**

5−x2x2+x−15-x2x2+x-1 =5−x2x2+2x−x−1=5-x2x2+2x-x-1 =5−x(2x(x+1)−1(x+1))=5-x(2x(x+1)-1(x+1)) =5−x(2x−1)(x+1)=5-x(2x-1)(x+1) Let 5−x2x2+x−1=A2x−1+Bx+15-x2x2+x-1=A2x-1+Bx+1 5−x2x2+x−1=A(x+1)+B(2x−1)(2x−1)(x+1)5-x2x2+x-1=A(x+1)+B(2x-1)(2x-1)(x+1)…

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**MATH**

**2×4+8×3+7×2−7x−12×3+4×2+4x2x4+8×3+7×2-7x-12×3+4×2+4x Write the partial fraction decomposition…**

You need to decompose the fraction in simple irreducible fractions, such that: 2×4+8×3+7×2−7x−12×3+4×2+4x=2×4+8×3+7×2−7x−12x(x2+4x+4)2×4+8×3+7×2-7x-12×3+4×2+4x=2×4+8×3+7×2-7x-12x(x2+4x+4)

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**MATH**

**x4+2×3+4×2+8x+2×3+2×2+xx4+2×3+4×2+8x+2×3+2×2+x Write the partial fraction decomposition of…**

x4+2×3+4×2+8x+2×3+2×2+xx4+2×3+4×2+8x+2×3+2×2+x on long division we get = > (x3+2×2+x)(x3+2×2+x) divides (x4+2×3+4×2+8x+2)(x4+2×3+4×2+8x+2) we get the Quotient’x’′x′ with remainder 3×2+8x+23×2+8x+2…

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**MATH**

**16×4(2x−1)316×4(2x-1)3 Write the partial fraction decomposition of the improper rational…**

16×4(2x−1)316×4(2x-1)3 Since the rational expression is an improper expression, we have to express the fraction as a sum of simpler fractions with the degree of the polynomial in the numerator less…

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**MATH**

**x4(x−1)3×4(x-1)3 Write the partial fraction decomposition of the improper rational expression.**

First make the substitution x−1=y,x-1=y, or x=y+1.x=y+1. Then the expression becomes (y+1)4y3.(y+1)4y3. Because (y+1)4=y4+4y3+6y2+4y+1,(y+1)4=y4+4y3+6y2+4y+1, the result is y+4+6y+4y2+1y3.y+4+6y+4y2+1y3. Recall y=x−1y=x-1 and…

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**MATH**

**x3+2×2−x+1×2+3x−4×3+2×2-x+1×2+3x-4 Write the partial fraction decomposition of the…**

x3+2×2−x+1×2+3x−4×3+2×2-x+1×2+3x-4 Since the rational expression is an improper expression , we have to express the expression as a sum of simpler fractions with the degree of the polynomial in the…

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**MATH**

**2×3−x2+x+5×2+3x+22×3-x2+x+5×2+3x+2 Write the partial fraction decomposition of the…**

You need to decompose the fraction in simple irreducible fractions, such that: 2×3−x2+x+5×2+3x+2=2×3−x2+x+5(x+1)(x+2)2×3-x2+x+5×2+3x+2=2×3-x2+x+5(x+1)(x+2)

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**MATH**

**x2−4xx2+x+6×2-4xx2+x+6 Write the partial fraction decomposition of the improper rational…**

x2−4xx2+x+6×2-4xx2+x+6 Rewrite the numerator of the rational expression as, x2−4xx2+x+6=x2+x+6−5x−6×2+x+6×2-4xx2+x+6=x2+x+6-5x-6×2+x+6 =(x2+x+6)−(5x+6)x2+x+6=(x2+x+6)-(5x+6)x2+x+6 =1−5x+6×2+x+6=1-5x+6×2+x+6 Polynomials do not…

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**MATH**

**x2−xx2+x+1×2-xx2+x+1 Write the partial fraction decomposition of the improper rational…**

x2−xx2+x+1×2-xx2+x+1 Rewrite the rational expression, x2−xx2+x+1=x2+x+1−2x−1×2+x+1×2-xx2+x+1=x2+x+1-2x-1×2+x+1 =(x2+x+1)−(2x+1)x2+x+1=(x2+x+1)-(2x+1)x2+x+1 =1−2x+1×2+x+1=1-2x+1×2+x+1 Polynomials do not completely divide and the…

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**MATH**

**x+1×3(x2+1)2x+1×3(x2+1)2 Write the partial fraction decomposition of the rational…**

You need to decompose the fraction in simple irreducible fractions, such that: x+1×3(x2+1)2=Ax+Bx2+Cx3+Dx+Ex2+1+Fx+G(x2+1)2x+1×3(x2+1)2=Ax+Bx2+Cx3+Dx+Ex2+1+Fx+G(x2+1)2 You need to bring to the…

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**MATH**

**8x−12×2(x2+2)28x-12×2(x2+2)2 Write the partial fraction decomposition of the rational…**

You need to decompose the fraction in simple irreducible fractions, such that: 8x−12×2(x2+2)2=Ax+Bx2+Cx+Dx2+2+Ex+F(x2+2)28x-12×2(x2+2)2=Ax+Bx2+Cx+Dx2+2+Ex+F(x2+2)2You need to bring to the same…

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**MATH**

**x2−4x+7(x+1)(x2−2x+3)x2-4x+7(x+1)(x2-2x+3) Write the partial fraction decomposition of the…**

x2−4x+7(x+1)(x2−2x+3)=Ax+1+Bx+Cx2−2x+3×2-4x+7(x+1)(x2-2x+3)=Ax+1+Bx+Cx2-2x+3 Multiply through by the LCD(x+1)(x2−2x+3)(x+1)(x2-2x+3) x2−4x+7=A(x2−2x+3)+(Bx+C)(x+1)x2-4x+7=A(x2-2x+3)+(Bx+C)(x+1) x2−4x+7=Ax2−2Ax+3A+Bx2+Cx+Bx+Cx2-4x+7=Ax2-2Ax+3A+Bx2+Cx+Bx+C…

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**MATH**

**x2+5(x+1)(x2−2x+3)x2+5(x+1)(x2-2x+3) Write the partial fraction decomposition of the rational…**

x2+5(x+1)(x2−2x+3)x2+5(x+1)(x2-2x+3) Letx2+5(x+1)(x2−2x+3)=Ax+1+Bx+Cx2−2x+3×2+5(x+1)(x2-2x+3)=Ax+1+Bx+Cx2-2x+3 x2+5(x+1)(x2−2x+3)=A(x2−2x+3)+(Bx+C)(x+1)(x+1)(x2−2x+3)x2+5(x+1)(x2-2x+3)=A(x2-2x+3)+(Bx+C)(x+1)(x+1)(x2-2x+3)…

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**MATH**

**3×4+x3x4+x Write the partial fraction decomposition of the rational expression. Check…**

3×4+x3x4+x Let’s factorize the denominator, x4+x=x(x3+1)x4+x=x(x3+1) =x(x+1)(x2−x+1)=x(x+1)(x2-x+1) Let 3×4+x=Ax+Bx+1+Cx+Dx2−x+13×4+x=Ax+Bx+1+Cx+Dx2-x+1…

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**MATH**

**x16x4−1x16x4-1 Write the partial fraction decomposition of the rational expression. Check…**

x16x4−1x16x4-1 Let’s factorize the denominator, 16×4−1=(4×2)2−116×4-1=(4×2)2-1 =(4×2+1)(4×2−1)=(4×2+1)(4×2-1) =(4×2+1)(2x+1)(2x−1)=(4×2+1)(2x+1)(2x-1) Let x16x4−1=A2x+1+B2x−1+Cx+D4x2+1x16x4-1=A2x+1+B2x-1+Cx+D4x2+1…

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**MATH**

**x2x4−2×2−8x2x4-2×2-8 Write the partial fraction decomposition of the rational expression….**

x2x4−2×2−8x2x4-2×2-8 Let’s factorize the denominator, x4−2×2−8=x4−4×2+2×2−8×4-2×2-8=x4-4×2+2×2-8 =x2(x2−4)+2(x2−4)=x2(x2-4)+2(x2-4)=(x2+2)(x2−4)=(x2+2)(x2-4) =(x2+2)(x+2)(x−2)=(x2+2)(x+2)(x-2) ∴x2x4−2×2−8=x2(x+2)(x−2)(x2+2)∴x2x4-2×2-8=x2(x+2)(x-2)(x2+2) Let…

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**MATH**

**2×2+x+8(x2+4)22×2+x+8(x2+4)2 Write the partial fraction decomposition of the rational…**

2×2+x+8(x2+4)22×2+x+8(x2+4)2 Let2x2+x+8(x2+4)2=Ax+Bx2+4+Cx+D(x2+4)22×2+x+8(x2+4)2=Ax+Bx2+4+Cx+D(x2+4)2 2×2+x+8(x2+4)2=(Ax+B)(x2+4)+Cx+D(x2+4)22×2+x+8(x2+4)2=(Ax+B)(x2+4)+Cx+D(x2+4)2 2×2+x+8(x2+4)2=Ax3+4Ax+Bx2+4B+Cx+D(x2+4)22×2+x+8(x2+4)2=Ax3+4Ax+Bx2+4B+Cx+D(x2+4)2…

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**MATH**

**x+6×3−3×2−4x+12x+6×3-3×2-4x+12 Write the partial fraction decomposition of the rational…**

x+6×3−3×2−4x+12x+6×3-3×2-4x+12 To decompose this to partial fractions, factor the denominator. x3−3×2−4x+12×3-3×2-4x+12 =(x3−3×2)+(−4x+12)=(x3-3×2)+(-4x+12) =x2(x−3)−4(x−3)=x2(x-3)-4(x-3) =(x−3)(x2−4)=(x-3)(x2-4) =(x−3)(x−2)(x+2)=(x-3)(x-2)(x+2)…

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**MATH**

**xx3−x2−2x+2xx3-x2-2x+2 Write the partial fraction decomposition of the rational…**

Decompose the denominator: x3−x2−2x+2=x2(x−1)−2(x−1)=(x2−2)(x−1)=(x−1)(x−2–√)(x+2–√).x3-x2-2x+2=x2(x-1)-2(x-1)=(x2-2)(x-1)=(x-1)(x-2)(x+2).Therefore the fraction decomposition has the form…

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**MATH**

**2xx3−12xx3-1 Write the partial fraction decomposition of the rational expression. Check…**

The denominator factors out as x3−1=(x−1)(x2+x+1),x3-1=(x-1)(x2+x+1),thus the general decomposition is 2xx3−1=Ax−1+Bx+Cx2+x+1.2xx3-1=Ax-1+Bx+Cx2+x+1. To find A, B and C, multiply both sides by x3−1:x3-1:

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**MATH**

**x2+2x+8(x2+4)2×2+2x+8(x2+4)2 Write the partial fraction decomposition of the rational…**

x2+2x+8(x2+4)2=x2+2x+8(x2+4)(x2+4)x2+2x+8(x2+4)2=x2+2x+8(x2+4)(x2+4) x2+2x+8(x2+4)2=Ax+Bx2+4+Cx+D(x2+4)2×2+2x+8(x2+4)2=Ax+Bx2+4+Cx+D(x2+4)2 Multiply through by the LCD (x2+4)2.(x2+4)2. x2+2x+8=(Ax+B)(x2+4)+(Cx+D)x2+2x+8=(Ax+B)(x2+4)+(Cx+D)…

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**MATH**

**6×2+1×2(x−1)26×2+1×2(x-1)2 Write the partial fraction decomposition of the rational…**

6×2+1×2(x−1)2=Ax+Bx2+Cx−1+D(x−1)26×2+1×2(x-1)2=Ax+Bx2+Cx-1+D(x-1)2 Multiply by the LCD x2(x−1)2.×2(x-1)2. 6×2+1=Ax(x−1)2+B(x−1)2+Cx2(x−1)+Dx26x2+1=Ax(x-1)2+B(x-1)2+Cx2(x-1)+Dx2 6×2+1=Ax(x2−2x+1)+B(x2−2x+1)+Cx3−Cx2+Dx26x2+1=Ax(x2-2x+1)+B(x2-2x+1)+Cx3-Cx2+Dx2…

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