Question

How do you do this problem?

Answer 1

Answer:

Step-by-step explanation:

You are going to work entirely on the right, until you can compare what is on the right with what you are given on the left.

a/1 + b/(x + 2) + c/(x - 8)

The first step is to determine the common denominator which is

(x + 2)(x - 8)

Now put each fraction over the common denominator.

a(x + 2)(x - 8) + b(x - 8) + c(x + 2)

             (x + 2)(x - 8)

The denominators on the left and right are the same, so you need only work with the numerators.

Remove the brackets.

a(x^2 - 6x - 16) + bx - 8b + cx + 2c

ax^2 - 6ax - 16a + bx - 8b + cx + 2c

In that whole expression, there is only 1 term that has an x^2 in it and that is the very first term.

a = 2 when you compare it to the left. You do this so you can get a value for  a.

Now put a = 2 in the first three terms on the expression above

2x^2 - 6*2x - 16(2) + bx - 8b + cx + 2c

2x^2 - 12x - 32 + bx - 8b + cx + 2c

The next thing is to get all the terms containing x together.

-12x + bx + cx = - 14x That's the term on the left (-14x). Divide by x

-12 + b + c = - 14          Add 12 to both sides

b + c = -14 + 12

b + c = - 2                    (x)

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Now you equate the terms with no x's in them.

-32 - 8b + 2c = - 76      Add 32 to both sides.

-8b + 2c = - 76 + 32

-8b + 2c = - 44             Divide by 2

-4b + c = - 22               (y)

Subtract (y) - (x)

-4b + c = - 22

  b + c = - 2

- 5b = - 20                   Divide by - 5

-5b/-5 = - 20/-5

b = 4

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b + c = - 2                     Find c substitute b = 4

4 + c = - 2                     Subtract 4 from both sides

c = - 2 - 4

c = - 6      

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Answer

a = 2

b = 4

c = -6