Multiply or divide, as indicated. (m^2 + 3m + 2)/(m^2 + 5m + 4) ÷...

Question

Multiply or divide, as indicated. (m^2 + 3m + 2)/(m^2 + 5m + 4) ÷ (m^2 + 5m + 6)/(m^2 + 10m + 24)

Accepted Answer

Hey everyone here we are asked to divide and simplify the given expression where we have a squared plus 13 A plus 42 all divided by a squared plus 15 A plus 54. And we divide this first fraction by our second fraction which has a numerator of a squared plus A plus 30 five. All divided by a squared plus 13 a plus 36. So now our first step here to divide this expression, we need to recall that when we divide by a fraction, we need to multiply by its reciprocal. So we need to rewrite our given expression where we instead multiply by the reciprocal of our second fraction. So our first fraction remains the same as a squared plus 13 A plus 42. All divided by a squared plus 15 A plus 54. And again we need to multiply by the reciprocal of our second fraction. So the numerator becomes a squared plus A plus 36. And this is all divided by the new denominator which is a square plus 12 A plus 35. So now our next step is to start simplifying and we need to do this by factoring this expression as much as possible. So we can start with factoring our first numerator here where we have a squared plus 13 A plus 42. Well the key to factoring here is to split our middle term into two terms. Which some to the 13 A but the product of the coefficients must be equal to 42. So our numerator becomes a squared plus six A plus seven A plus 42. And we see that six A plus seven A. Works because they some 2 13 A but six times seven does equal 42. So now we can repeat this process with the rest of our expression. We can now move on to the denominator of our first fraction here. So we see that the middle term is 15 a. And we see that the product must be equal to 54. So our denominator becomes a squared plus nine a plus six A plus 54. And again we see that nine A plus six A. Does equal 15 A. But nine times six does also equal 54. So now we can move on to our second fraction here in the numerator we have a squared plus 13 A plus 36. And so we can split the middle term of 13 A. Into nine A plus four A. So our numerator becomes a squared plus nine A plus four A plus 36. And now we can move on to the denominator where our middle term is 12 A. And so our denominator becomes a squared plus five A. Plus seven A plus 35. So now we just need to use the coefficient of our middle terms to finish writing the factored form of the numerator is and the denominators. So again beginning with our first numerator we see that one of our coefficients is six and the other coefficient is seven. So the factored form of our first numerator becomes the quantity of a plus six times the quantity of a plus seven. And now moving on to our denominator, we see that one of our coefficients here is nine and the other coefficient is six. So our factored form becomes the quantity of a plus nine times the quantity of a plus six. And now we can move on to our second fraction. So again looking at the numerator, we see that one coefficient is nine and the other coefficient is four. So the factored form of our second numerator is the quantity of a plus nine times the quantity of a plus four. Moving on to the denominator of our second fraction. We see that one coefficient is five and the other coefficient is seven. So the factored form becomes the quantity of a plus five times the quantity of a plus seven. So with this we have fully factored our expression so we can now start making simplifications by canceling out the same terms in the numerator and the denominators. So here we can see that in the numerator of our first fraction we have the quantity of a plus six. Well we also have the quantity of a plus six in the denominator, so we can cancel out both of these terms. And then we also see that in the denominator of our first fraction we have the quantity of a plus nine which also appears in the numerator of our second fraction. So we can also cancel both of these terms. And now we also see in the numerator of our first fraction we have the quantity of A plus seven which also appears in the denominator of our second fraction, so both of these terms cancel out. So now we can rewrite our expression with the terms we have left and we see we are left with the quantity of a plus four divided by the quantity of a plus five. And with this we have fully simplified our original expression and therefore we are left with answer choice B. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Multiply or divide, as indicated. (m^2 + 3m + 2)/(m^2 + 5m + 4) ÷ (m^2 + 5m + 6)/(m^2 + 10m + 24)

Key Concepts

Rational Expressions

Factoring Polynomials

Division of Fractions

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