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Simplify the function ๐ of ๐ฅ is equal to ๐ฅ plus five divided by ๐ฅ squared plus nine ๐ฅ plus 20 times ๐ฅ squared plus 15๐ฅ plus 54 divided by seven ๐ฅ squared plus 69๐ฅ plus 54 and determinate its domain.
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Essentially, we need to take each expression and simplify it by factoring.
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The ๐ฅ plus five does not factor, so itโs good.
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However, the one below it, what two numbers multiply to be 20 and add to be nine?
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That would be five and four.
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So we can replace that with ๐ฅ plus five times ๐ฅ plus four.
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In the top right corner, what are two numbers that multiply to be 54 and add to be 15?
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That would be nine and six.
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Both of those were simple trinomials; however, in the bottom right-hand corner, this is an advanced trinomial because the leading coefficient of seven isnโt one.
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So weโre going to use the slip and slide method.
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So letโs go ahead and slip this seven to the back by multiplying it to 54.
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So now we have ๐ฅ squared plus 69๐ฅ plus 378.
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So what two numbers multiply to be 378 and add to be 69?
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This might take a little bit, but eventually we will come up with the numbers, and they are 63 and six.
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Now, the number that we slipped to the back, we are now going to have two slide underneath and simplify.
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So 63 divided by seven is nine.
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Now, six-sevenths, that doesnโt reduce, so Iโm going to bring this seven up with the ๐ฅ.
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When factoring, we donโt usually leave fractions.
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So now, our last expression has been replaced.
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When simplifying or multiplying two fractions, we can cancel things up and down and diagonally.
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So the ๐ฅ plus fives cancel, the ๐ฅ plus nines cancel, and thatโs it!
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So the only thing left on the top is the ๐ฅ plus six and on the denominator, thereโs an ๐ฅ plus four and seven ๐ฅ plus six.
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Now, we have to figure out our domain.
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So our domain is anything, except for what will make the denominator zero.
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So we need to set every factor that was on the bottom equal to zero.
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And after doing that, if you set ๐ฅ plus five equal to zero, you get negative five.
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And then we get negative four, we get negative nine, and negative six-sevenths.
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So the domain will be all real numbers minus negative nine, negative five, negative four, and negative six-sevenths.
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So our function simplified is ๐ of ๐ฅ equals ๐ฅ plus six divided by ๐ฅ plus four times seven ๐ฅ plus six, with a domain is all real numbers minus negative nine, negative five, negative four, and negative six-sevenths.