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Given that the algebraic fraction π of π₯ equals eight π₯ times π₯ plus four over π₯ plus π simplifies to π of π₯ equals eight π₯, what is the value of π?
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In order to answer this question, weβre going to need to recognize how we simplify an algebraic fraction.
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To simplify an algebraic fraction, we need to look for shared or common factors in the numerator and denominator of that fraction.
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When weβve identified those, we know that we can cancel through by this shared factor, leaving a much-simpler expression.
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Take the function π of π₯ which is eight π₯ times π₯ plus four over π₯ plus π.
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Since this fraction simplifies to eight π₯, it must be equal to eight π₯ for all values of π₯.
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Now, in fact, on the left-hand side, we do have eight π₯ already.
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So, this tells us that we should be able to cancel by dividing through by a common factor with the remaining terms.
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In turn, this means that the expression π₯ plus four must be equal to π₯ plus π for all values of π₯.
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And, of course, the only way that this can be true is if π itself is equal to four.
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And so, given that the algebraic fraction simplifies to eight π₯, we can deduce that the value of π is four.