<p>So the section of the chapter is "Partial Fractions", the question is "write the form of the partial fraction decomposition of the rational expression. Then solve for the constancts"
(4x^2 + 3)/(x + 5)^3</p>
<p>then i got it to:A/(x-5) + B/(x-5)^2 + C/(x-5)^3
but i dont know how to solve for the constants... help! hope this makes sense</p>
<p>Alright first off you wrote "x+5" in your original question but "x-5" for each factor when you split them up... I'll assume that it's "x+5" and show you how to solve that.</p>
<p>First we have what you've already done:
(4x^2 + 3)/(x + 5)^3 = A/(x+5) + B/(x+5)^2 + C/(x+5)^3</p>
<p>Multiply both sides through by (x+5)^3 to get rid of the denominator on the left so that it is a quadratic. Now we have:</p>
<p>4x^2 + 3 = A(x+5)^2 + B(x+5) + C</p>
<p>Multiply everything through on the right side, so you end up with:</p>
<p>4x^2 + 3 = Ax^2 + 10Ax + 25A + Bx + 5B + C</p>
<p>Now, factor like terms on the right side so that the x^2's are grouped together, the x's are grouped together, like in a quadratic equation, etc:</p>
<p>4x^2 + 3 = Ax^2 + (10A + B)x + (25A + 5B + C)</p>
<p>Now the two sides look very familiar, right? Now you just need to equate the coefficients for each term. In other words, look at the x^2 term on both sides. You get that A=4. Next look at the x term on each side. On the left, it is "0", so set (10A + B) = 0. We just figured out that A=4, so substitute that in and solve to find that B=-40. Finally, look at the "constant" term on both sides. On the left it's 3 and on the right it's (25A + 5B + C). Knowing that A is 4 and B is -40, solve to find that C=103.</p>
<p>So putting those back in (A=4, B=-40, C=103) you have:
(4x^2 + 3)/(x + 5)^3 = 4/(x+5) - 40/(x+5)^2 + 103/(x+5)^3</p>
<p>thats so much! i get it!</p>