Question

Question 6

Which of the following is equivalent to the expression
denominator does not equal zero.)

Answer 1

Choice 1

`6x + 13 - (7x - 99)/(x^2 - 3)`

Explanation:

The reasonable way to solve this is to multiply each of the expressions in the choices by the denominator and see which resulting expression in the numerator corresponds to the original numerator

We can eliminate the last two choices since denominator of the second expression in both has a highest degree of x as 3 (x³) and multiplying x³ by 6x will result in a degree of 4 which is not in the numerator of the original expression

So let's try the first choice:

`6x + 13 - (7x - 99)/(x^2 - 3)`

Multiplying throughout by
`x^2 - 3` gives:

`(6x + 13)(x^2 - 3) - (7x-99)/(x^2 - 3) \cdot (x^2 - 3)\\\\\implies (6x + 13)(x^2 - 3) - (7x - 99)\\\\\implies (6x + 13)(x^2 - 3) -7x + 99\cdots\cdots(1)\\\\`

`(6x + 13)(x^2 - 3) = (6x + 13)x^2 + (6x + 13)(-3)\\\\= 6x^3 + 13x^2 - 18x - 39`

So expression (1) becomes:

`6x^3 + 13x^2 - 18x -39 -7x + 99\\\`

Grouping like terms -18x and -7x we get

`6x^3 + 13x^2 - 18x - 7x - 39 + 99\\\\\rightarrow 6x^3 + 13x^2 - 25x + 60\\\\`

This corresponds to the numerator of the original expression and therefore the correct choice is the first choice

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If we wanted to check the other choice (not necessary) you would find that the expression when multiplied by (x² - 3) will become

`(6x + 13)(x^2 - 3) + 7x - 99\\\\\text{which becomes}\\\\6x^3 + 13x^2 - 11x - 138`

which does not correspond to the numerator of the given expression