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CHALLENGE Divide $ rac{3}{4}$ by $ rac{1}{2}, rac{1}{4}, rac{1}{8},$ and $ rac{1}{12} .$ What happens to the quotient as the value of the divisor decreases? Make ...

Question

CHALLENGE Divide $ rac{3}{4}$ by $ rac{1}{2}, rac{1}{4}, rac{1}{8},$ and $ rac{1}{12} .$ What happens to the quotient as the value of the divisor decreases? Make a conjecture about the quotient when you divide $ rac{3}{4}$ by fractions that increase in value. Test your conjecture.

CHALLENGE Divide $\frac{3}{4}$ by $\frac{1}{2}, \frac{1}{4}, \frac{1}{8},$ and $\frac{1}{12} .$ What happens to the quotient as the value of the divisor decreases? Make a conjecture about the quotient when you divide $\frac{3}{4}$ by fractions that increase in value. Test your conjecture.



Answers

Find the quotient and the remainder. Check your work by verifying that (Quotient) (Divisor) $+$ Remainder $=$ Dividend $$ -4 x^{3}+x^{2}-4 \text { divided by } x-1 $$

So for this question, we are dividing native tricks. The fourth minus two X minus one by X minus one. So one thing to know here is whenever good doing polynomial long division and its comes in the form where you have like an X to the fourth. We don't have an X cubed or an X squared term. You need to still fill these in with a coefficient of zero to make the long division work. And you'll see why in this problem. So when we're dividing negative drinks, the fourth by X will start off with that. How many times does Exco Internet? But three is to the fourth? Well, it would be negative. Three Ex cute. So when we multiply, we first of all we have toe subtract the multiplication of native threats cubed times acts which would be the negative three extra fourth. Then we also have to consider the one what is negative three x cubed times, one negative one. So that would be plus three x cubed. And then what we have to do is subtract, thes and carry down everything that doesn't get subtracted. So this turned into a zero. This would turn into the negative three x cubed and then all other terms get carried down. So we carried down the zero x squared, the negative two X and the negative one. So next how many times does X go into negative three x Cubed? Well, if it went into negative search, the fourth negatives reacts Q times. It would go into negative tricks. Cube. Negative three X squared times cause native three X squared Times X would be the negative three x cubed. So let's do the same step We did this before, where we have to subtract. And here we have to subtract negative three x cubed and then whatever negative one times three X squared is which would be three x squared. So the next step carry everything down that can be carried down. Disturbed into a zero. This would be subtracting this three square. It's a three egg native three x squared carried on this negative to eggs and carried down the negative one. How many times does this ex term now go into negative to re X squared Well by pattern, we fear that out. It would be negative. Three x times native three x times x would be the negative three x squared, then negative one times negative. Three x were adding this three x term. So one last the vision and what we gay Here we get this turned into zero negative. Two X minus three x negative five X minus one. Okay, let's keep going. So x times what will become negative? Five x It would be negative. Five. When we do that, you can see that. Let's just carry all this up here. So we say we have this negative five experts one and we're now subtracting the negative five Times X, which would be plus five X and the negative five times negative one, which would be plus five. Let me do that. This turns into zero and this last term would turn into negative six. So that's a remainder. Because we can't divide anymore Just to validate that we have this negative three x cubed times X minus one plus six would equal this negative three x to the fourth minus two x minus one. So let's just make sure of that when we multiply x times, maybe three x cubed we get three X could have or the *** three to the fourth. We want to play X by neighbor three X squared. We get negative. Three extra. Okay, X times negative three x will give us near three X squared x Times number five will give us *** five x. Okay, now negative one times three x cubed will give us just three execute. So just we're adding another three. Execute here. Negative one times needed. Three X squared will give us three X squared native one times negative five Sorry, negative. One times negative three will give us a plus three x and then a negative one times in eight if I will give us plus five. So what? We eliminate some terms that we can you can eliminate thes thes. So then what comes down is our negative three x to the fourth and negative five X plus three X minus two X plus five. But then we also have to consider our remainder over here. So plus five minus six. So we get by this one which, when we look at it air original polynomial was negative tricks of the fourth minus two x minus one. So this all checks out. So when we divide negative threats of fourth minus two X minus one by X minus one We get native three x cubed minus three three X squared minus three X minus five, with the remainder of negative six.

The key point to remember here is that if you're dividing by a fraction, it's equivalent to multiplying by its reciprocal. So three divided by 1/4 is equivalent to three multiplied by 4/1. That's just the same as three times for, and so we would get 12.

We're being asked to divide one and 1/2 by three fours. Well, remember, when you divide fractions, you can't have that next number. So first we have to change one and 1/2 into an improper fraction. To do this, we divide or sorry, we multiply the denominator by the whole number and then add the new murder. So two times one is two plus one is three and you're denominated will stay to say so. That will be the Fraction 3/2. So now we have to divide 3/2 by three fours. Well, to divide fractions, we multiplied by the reciprocal. So we'll have 3/2 times, while the reciprocal of 3/4 is for over three. So now I can multiply straight across. Well, three times four is 12 and two times three is six. And now we have 12 divided by six, which is just too. So too, is our final answer.

We wish to simplify this quotient here. So let's first rewrite this as a multiplication. You look copy down the first expression and then changed to multiple patients that were multiplying by the reciprocal. So that means the second expression. So now we can go ahead and factor everything that we can. The only disseminator can get factory. I got a three and will be much. But experts for and now we can cancel out factors that appear on top and bottom four elevating acts and also the two. And the four can get simplified into one and two analogy. Multiply straight across on top. We have three times one, which is just three. And on the bottom we have two times one nutritious.


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