Can you help me work this out with long division?
r^2 - 7r + 12/ r+4 divided by r^2 - 7r +12/ r^2 + 6r +8
Thanks.
r^2 + 6r +8 = (r+2)(r+4)
So, what you have is
(r^2-7r+12) / (r+4) * (r+2)(r+4) / (r^2-7r+12)
Now lots of stuff cancels out and you are left with just
(r+2)
Of course, I can help you with long division. To divide fractions, we need to follow a step-by-step process. Let's break it down:
Step 1: Write the problem in division form.
We have: [(r^2 - 7r + 12)/(r + 4)] ÷ [(r^2 - 7r + 12)/(r^2 + 6r + 8)]
Step 2: Keep the first fraction as it is and convert the division sign to multiplication. Then, flip the second fraction (this is known as "taking the reciprocal").
[(r^2 - 7r + 12)/(r + 4)] × [(r^2 + 6r + 8)/(r^2 - 7r + 12)]
Step 3: Factorize the numerator and denominator of each fraction.
Numerator of the first fraction: r^2 - 7r + 12 = (r - 3)(r - 4)
Denominator of the first fraction: r + 4
Numerator of the second fraction: r^2 + 6r + 8 = (r + 2)(r + 4)
Denominator of the second fraction: r^2 - 7r + 12 = (r - 3)(r - 4)
Now, let's rewrite the expression:
[(r - 3)(r - 4)/(r + 4)] × [(r + 2)(r + 4)/((r - 3)(r - 4))]
Step 4: Cancel out any common factors between the numerators and denominators.
In this case, we can cancel out (r - 3) and (r - 4):
[(r - 3) × (r - 4)/(r + 4)] × [(r + 2)(r + 4)/((r - 3)(r - 4))]
Step 5: Multiply the remaining factors.
(r - 4) and (r + 4) cancel out, leaving us with:
[(r - 3)/(r + 4)] × [(r + 2)/(r - 3)]
Step 6: Multiply the resulting fractions together.
(r - 3) and (r - 3) cancel out, leaving us with:
(r + 2)/(r + 4)
Therefore, the final result is:
(r + 2)/(r + 4)