Year 13 Maths Partial fractions: Repeated Linear factors

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EdBedd
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4x²+16x+15/(x+1)(x+2)²= A/(x+1) + B/(x+2)¹ + C/(x+2)²
My questions is that why is it this
4x²+16x+15= A(x+2)² + B(x+1)(x+2) + C(x+1)
and not this
4x²+16x+15= A(x+2)² + B(x+1)(x+2) + C(x+1) (x+2)
I'm not sure where the extra (x+2), i've been told you cancel out the (x+2)² with a (x+2)² leaving just (x+1) but why does that happen?

Sorry if this question is bad, first time posting lol
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MindMax2000
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(Original post by EdBedd)
4x²+16x+15/(x+1)(x+2)²= A/(x+1) + B/(x+2)¹ + C/(x+2)²
My questions is that why is it this
4x²+16x+15= A(x+2)² + B(x+1)(x+2) + C(x+1)
and not this
4x²+16x+15= A(x+2)² + B(x+1)(x+2) + C(x+1) (x+2)
I'm not sure where the extra (x+2), i've been told you cancel out the (x+2)² with a (x+2)² leaving just (x+1) but why does that happen?

Sorry if this question is bad, first time posting lol
In partial fractions, you're multiplying the numerator with the factor missing from the denominator.
4x²+16x+15= A(x+2)² + B(x+1)(x+2) + C(x+1) makes sense because C has (x+2)^2, so it only needs to be multiplied by (x+1). (I also can't read the indices, since the font is too small.)
If you take the example of x =1:
A/(x+1) + B/(x+2)¹ + C/(x+2)² = A/2 + B/3 + C/9
You would need A(9) + B(6) + C(2) in order to put it all over 18 in the fraction. C(6) would mess up the values in the equation.
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EdBedd
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(Original post by MindMax2000)
In partial fractions, you're multiplying the numerator with the factor missing from the denominator.
4x²+16x+15= A(x+2)² + B(x+1)(x+2) + C(x+1) makes sense because C has (x+2)^2, so it only needs to be multiplied by (x+1). (I also can't read the indices, since the font is too small.)
If you take the example of x =1:
A/(x+1) + B/(x+2)¹ + C/(x+2)² = A/2 + B/3 + C/9
You would need A(9) + B(6) + C(2) in order to put it all over 18 in the fraction. C(6) would mess up the values in the equation.
Thank you, i understand it now. Actually listing out what you multiply the numerator made me realise that the C(x+1) does have the other (x+2) canceled out and now i see why it goes.
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