Prove the function : f(x) = [(2x+1)(x+4)] /√x , can be written in the form: Px^(3/2)+ Qx^(1/2) + Rx^(-1/2)
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Prove the function : f(x) = [(2x+1)(x+4)] /√x , can be written in the form: Px^(3/2)+ Qx^(1/2) + Rx^(-1/2)

[From: ] [author: ] [Date: 11-12-24] [Hit: ]
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Please explain all your steps. Thanks.

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f(x) = [(2x+1)(x+4)] /√x => multiply the parenthesis (numerator):
= (2x^2 + 8x + x + 4) / √x => simplify(num):
= (2x^2 + 9x + 4) / √x => rationalize the denominator:
= (2x^2 + 9x + 4) / √x * [√x/√x]
= [√x * (2x^2 + 9x + 4)] / [√x * √x] => √x = x^(1/2):
= [(2x^2 * x^(1/2) + 9x * x^(1/2) + 4 * x^(1/2)] / x => 1/x = x^(-1)
= [2x^2 * x^(1/2) * x^(-1) + 9x * x^(1/2) * x^(-1) + 4 * x^(1/2) * x^(-1)]
using rules of exponents:
= [2 * x^(2 + 1/2 -1) + 9 * x^(1 + 1/2 -1) + 4 * x^(1/2 - 1)]
f(x) = 2 * x^(3/2) + 9 * x^(1/2) + 4 * x^(-1/2)
f(x) = P * x^(3/2) + Q * x^(1/2) + R * x^(-1/2)
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