Answer:
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