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I'm working on the Simplify Rational Expressions with Common Binomial Factors (https://www.khanacademy.org/math/algebra2/rational-expressions-equations-and-functions/simplify-rational-expressions/e/simplifying_rational_expressions_2) on Khan Academy. One of the questions has me a little stumped. The original question is this:

Algebraic Fraction Question

After simplifying it, you get to this situation:

Opposite Negatives

As far as I understand, the hints tell me to multiply the numerator by -1 and distribute that negative one to the factor which is similar but opposite to the factor in the denominator. In this case the factor in the numerator is (k-14) and the one in the denominator is (14-k).

By multiplying the numerator by -1, the opposite factor becomes (14-k) and we can cancel the two factors out.

The part of the question that has me stumped is right at the end. In the hints, Sal Khan seems to just drop a negative sign from the fraction, and I can't see any mathematical mechanics that allowed him to do so. I clearly haven't understood the full implications of multiplying the numerator by -1 the first place.

Confusing Part

I look forward to hearing some explanations about how this all works.

Thank you in advance for your time.

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If you have a minus sign in front of a fraction, you can remove it by multiplying it into the denominator (or even the numerator in fact). That is, the following is true: $$-\frac{a}{b} = \frac{a}{-b}.$$

Therefore, we have $$-\frac{a}{z-5} = \frac{a}{-(z-5)} = \frac{a}{5-z}.$$

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    $\begingroup$ Thank you. I was concerned it was something more complex. What is the dot above the b and z? $\endgroup$
    – Owen Alexander
    Commented Feb 20, 2019 at 5:14
  • $\begingroup$ That dot is just a full stop to finish the sentence, nothing mathematical. $\endgroup$
    – Minus One-Twelfth
    Commented Feb 20, 2019 at 5:17
  • $\begingroup$ ah ok thank you :) $\endgroup$
    – Owen Alexander
    Commented Feb 20, 2019 at 5:24

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