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Show that negative four 𝑥 minus 12 over 𝑥 minus four divided by 𝑥 squared plus seven 𝑥 plus 12 over 𝑥 cubed minus 16𝑥 simplifies to 𝑎𝑥, where 𝑎 is an integer.
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Our first step here is to consider what happens when we divide by a fraction.
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Well, 𝑎 over 𝑏 divided by 𝑐 over 𝑑 can be rewritten as 𝑎 over 𝑏 multiplied by 𝑑 over 𝑐.
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We can multiply by the reciprocal of the second fraction.
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In this case, our question can be rewritten as negative four 𝑥 minus 12 over 𝑥 minus four multiplied by 𝑥 cubed minus 16𝑥 over 𝑥 squared plus seven 𝑥 plus 12.
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We now need to simplify the numerators and denominators by factorizing.
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Negative four 𝑥 minus 12 can be simplified to negative four multiplied by 𝑥 plus three by factorizing out negative four.
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This is because negative four multiplied by 𝑥 is negative four 𝑥 and negative four multiplied by three is negative 12.
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The numerator of the second fraction 𝑥 cubed minus 16𝑥 has 𝑥 as a common factor.
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Therefore, it can be rewritten as 𝑥 multiplied by 𝑥 squared minus 16.
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This bracket in turn can be factorized again using the difference of two squares.
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This gives us 𝑥 multiplied by 𝑥 plus four multiplied by 𝑥 minus four.
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Finally, we can factorize the denominator of the second fraction into two brackets.
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The first term of these brackets will be 𝑥.
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And the second terms need to have a product of 12 and a sum of seven.
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Three multiplied by four is equal to 12 and three plus four equals seven.
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Therefore, our two brackets are 𝑥 plus three and 𝑥 plus four.
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We can therefore rewrite our expression as negative four multiplied by 𝑥 plus three over 𝑥 minus four multiplied by 𝑥 multiplied by 𝑥 plus four multiplied by 𝑥 minus four over 𝑥 plus three multiplied by 𝑥 plus four.
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Our next step is to try to cancel brackets that appear on the top and the bottom of the fractions.
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Firstly, we can cancel 𝑥 plus four in the second fraction.
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We can also cancel an 𝑥 plus three from the top of the first fraction and the bottom of the second fraction.
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Finally, we can cancel 𝑥 minus four from the top of the second fraction and the bottom of the first fraction.
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This leaves us with minus four multiplied by 𝑥, which we can simplify to negative four 𝑥.
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Our expression has therefore been simplified to the form 𝑎𝑥, where 𝑎 is an integer, in this case negative four.