When 2x^2−7x+9 is divided by a polynomial, the quotient is 2x-3 a...

Question

When 2x^2−7x+9 is divided by a polynomial, the quotient is 2x-3 and the remainder is 3. Find the polynomial.

Accepted Answer

Welcome back. I'm so glad you're here. We are to find a polynomial. And were given a lot of information about it. We're told that three X squared plus X plus four yields a quotient of three X -1 and a remainder of eight when divided by this polynomial. So this is how we set up the equation. A real quick review that when we've got a remainder that it's going to be divided by the original divisor, which is what we're looking for here. The divisor is the thing that the dividend up here is being divided by. So we've set it up this way and now we're just trying to find peace trying to find that polynomial. We can solve this algebraic lee because P is a denominator here, it's a common denominator. We can multiply all of these by P. And on the left hand side the peas will cancel out. We'll just have three X squared plus 11 X plus four. And on the right hand side will have a three x minus one in parentheses, times P. And then for that plus eight over P the pi's cancel out. And so it's just plus eight. And we are trying to solve for P. I'll keep that in blue. Trying to solve for P. So let's subtract eight from both sides on the left we'll have three X squared plus 11 X minus four equals. And on the right hand side it's still p times parentheses, three x minus one. Now we can divide both sides by three X -1. And on the right we've just got P. And on the left we've got three X squared plus 11 X minus four divided by three x minus one. Alright now we just need to simplify on the left hand side. We could do the division, we can try and factor that numerator. See if there's anything that will cancel out. And we could see if we have a factor of three X -1 here. So looking at that three X square, that's going to be a three X times an X to get three X squared. So this could be a three x minus one. Let's see if there is a term, we can stick over here with this X to get three X squared plus 11 X minus four. Well, to multiply to get a negative four, you need a negative one times a positive four. And let's check that real quick. Let's foil it out to see if it works three x minus one times X plus four. Our first three x times X is three X squared. Our Outsides three X times four is plus 12 X. Our insides negative one times X is negative one X. That looks good. And our Outsides negative one times four is a negative for combining our inner like terms we have 12 x minus one, X is plus 11 X. And then bring down our minus four. Yes, this matches what we had originally. So we can factor this out. Alright, this over here, our numerator is the factor three X minus one times X plus four. And our denominator is that three x minus one. R three X minus one factors cancel out, and P just equals X plus four. We look at our answer choices in this matches with answer choice, a well done, we'll catch you on the next one.

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When 2x^2−7x+9 is divided by a polynomial, the quotient is 2x-3 and the remainder is 3. Find the polynomial.