Describe and correct the error.
`((r^2 - 7r + 12)/(r + 4)) / ((r^2 - 7r + 12)/(r^2 + 6r+ 8)) = (((r-3)(r-4))/(r+4)) / (((r-4)(r-3))/((r+2)(r+4))) = (r+4)/((r-3)(r-4)) * ((r-4)(r-3))/((r+2)(r+4)) = 1/(r+2)`
The error here is that the second expression is not equal to the third.
When dividing rational expressions (fractions), one needs to multiply the FIRST fraction by the reciprocal of the SECOND fraction. Instead, here the reciprocal of the FIRST fraction is multiplied by the SECOND fraction.
The correct answer should be the reciprocal of the answer obtained here.
This is the correct way to solve this division problem:
`((r^2 - 7r + 12)/(r+ 4)) / ((r^2 - 7r + 12)/(r^2 + 6r + 8)) = (r^2 - 7r + 12)/(r + 4) * (r^2+6r + 8)/(r^2 - 7r + 12) `
`` The numerator of the first and the denominator of the second fraction cancel each other. The expression equals
`(r^2 + 6r +8)/(r + 4) = ((r+4)(r + 2))/(r + 4) = r + 2`
The answer is r + 2, which is the reciprocal of the original incorrect answer, `1/(r + 2)` , as expected.