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How do you divide #\frac { 3k ^ { 2} - 2k - 1} { 3k ^ { 2} + 10k + 7} \div \frac { 9k ^ { 2} - 1} { 3k ^ { 2} + 4k - 7}#?

Answer 1

Isaiah Anthony

#frac[(k-1)^2][(k+1)(3k-1)]#

#frac(3k^2-2k-1)(3k^2+10k+7) -: frac(9k^2-1)(3k^2+4k-7)#

To divide fractions, multiply by the 1st by the reciprocal of the 2nd.

#frac(3k^2-2k-1)(3k^2+10k+7) * frac(3k^2+4k-7)(9k^2-1)#

Factor both numerators and denominators.

#frac[(3k+1)(k-1)][(3k+7)(k+1)] * frac[(3k+7)(k-1)][(3k+1)(3k-1)]#

Cancel factors found in both numerator and denominator.

#frac[color(red)cancel((3k+1))(k-1)][color(blue)cancel((3k+7))(k+1)] * frac[color(blue)cancel((3k+7))(k-1)][color(red)cancel((3k+1))(3k-1)]#

Multiply the remaining factors straight across.

#frac[(k-1)(k-1)][(k+1)(3k-1)]#

The numerator can be rewritten as th square of the binomial.

#frac[(k-1)^2][(k+1)(3k-1)]#

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.