Multiply or divide, as indicated. ac + ad + bc + bd/a²

Question

Multiply or divide, as indicated. ac + ad + bc + bd/a² - b² * a³ - b³/2a² + 2ab + 2b²

Accepted Answer

Hey everyone here we are asked to simplify the following expression by applying the operations indicated. So here our given expression we have em you plus two M v minus n. U minus two envy. And this is all divided by the quantity of m squared minus n squared. And now this first fraction is multiplied by another fraction which has the numerator of M cubed plus n cubed. And this is all divided by the quantity of five, M squared minus five M n plus five and squared. So now our first step here in simplifying this given expression is to just factor out common terms in the numerator and nominators. So starting with our first fraction looking at the numerator. Well in the first two terms we have em you plus two mv. Well the common term here in these two terms is M so we can go ahead and factor it out and this gives us m times the quantity of U plus two V. And now for our last two terms we have minus and you minus two envy. So here we both have both of these terms have a minus sign and they also both contain n. So we can go ahead and factor out minus n. And this is times the quantity of U plus two V. And now our denominator here for our first fraction is just going to remain the same for now as m squared minus m squared. And now this fraction we can look at the second fraction again we're going to keep the numerator the same for now as m cubed plus n cubed. But now looking at our denominator of our second fraction, we see that each of these terms here we have five M squared minus five M n plus five N squared. Well, again, each of these terms here have a five, so we can factor out five, giving us five times the quantity of m squared minus M N plus n squared. So now we have finished doing some simple factory in here but we can continue to factor this expression further by first noticing that in the deny dominator of our first fraction here we have the difference of two squares and squared minus and squared. So we want to factor this expression and we can do this by recalling the difference of squares formula which tells us that a squared minus B squared is equal to the quantity of a plus B times the quantity of a minus B. So now looking at our expression in the denominator, we see that our value for a is going to be M and our value for B is going to be n. So now we can go ahead and rewrite m squared minus n squared according to our formula here, which tells us that it will factor as the quantity of M plus n times the quantity of m minus n. And now we can look at our second fraction here looking at the numerator, we see that we can factor this expression further because we have the sum of two cubes M cubed plus n cubed. So to factor this expression we can recall the sum of two cubes formula which tells us that X cubed plus y cubed is equivalent to the quantity of X plus Y times the quantity of X squared minus x, Y plus y squared. And so now comparing this formula with our second fractions numerator, we see that our value for X is going to be M and our value for Y is going to be N. So now we can go ahead and use our formula to rewrite and cubed plus N cubed as it's factored form which becomes the quantity of M plus N times the quantity of M squared minus M times n plus n squared. And so now we have found the factored form of the denominator of our first fraction as well as the numerator of our second. So we can go ahead and input this new information into our expression. So rewriting our expression again in the numerator we actually can also perform another simplification where we group together the m minus N. So we have the quantity of m minus n times the like term here which is this quantity of U. Plus to V. And now for our denominator we just need to include what we found earlier because we see that our difference of squares becomes the quantity of M plus n times the quantity of M minus n. And now looking at our second fraction here we see our numerator can we can be replaced with our sum of cubes formula or what we found from our formula. So it becomes the quantity of M plus n times the quantity of M squared minus M times n plus n squared. And the denominator remains the same. So this is all divided by the denominator where we have five times the quantity of M squared minus M times N plus n squared. So now that we have fully factored our original expression, we can go ahead and start making some simplifications by canceling the terms that are ending numerator and the denominator. So beginning with our first fraction here we see that in the numerator we have this quantity of m minus n. And we also have this term in the denominator. So we can go ahead and cancel out these 22 terms. And now looking at our second fraction here, we see that we have this quantity of m squared minus M times N plus and squared. And this expression also appears in the denominator so we can go ahead and cancel both of these terms out. And so now since we are multiplying these two fractions together, we can also simplify the the like enumerators, the like terms and enumerators with the like terms in the denominators in both of the fractions. So we see in the numerator of the second fraction we have this quantity of M plus n. And this also appears in the denominator of our first fraction. So again, because we're multiplying the fractions together, we can cancel these terms out. So now that we have made all all of these simplifications that we can make, we can go ahead and rewrite our expression with the remaining terms that we have. And we see that we have the quantity of U plus two V, all divided by five. And this is our final simplified answer, which is 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. ac + ad + bc + bd/a² - b² * a³ - b³/2a² + 2ab + 2b²

Key Concepts

Polynomial Multiplication and Distribution

Factoring Polynomials

Simplifying Rational Expressions

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Multiply or divide, as indicated. ac + ad + bc + bd/a2 - b2 * a3 - b3/2a2 + 2ab + 2b2

Key Concepts

Polynomial Multiplication and Distribution

Factoring Polynomials

Simplifying Rational Expressions

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Multiply or divide, as indicated. ac + ad + bc + bd/a² - b² * a³ - b³/2a² + 2ab + 2b²