Solve the equation 12x^3+16x^2−5x−3=0 given that -3/2 is a root.

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Accepted Answer

Welcome back. I am so glad you're here. We are given this equation 12 X cubed minus 35 X squared plus 28 X minus five equals zero. We're told that 5/3 is one of the roots of this equation. And then we're to solve for the other X values that make this true equal to zero. So we're to find the other routes. Well to find those other routes, we know that 5/3 is going to work. I recommend starting with synthetic division using five thirds will bring down all of the coefficients in the constant. For synthetic divisions we've got 12 negative positive 28 negative five. And then we'll go ahead and do synthetic division to find our new polynomial equation. And then we'll see if we can factor once we have X squared as our highest degree. Okay, so the first step in synthetic division we recall from previous lessons, this first step only happens the first time we're just going to bring down the first term. So bringing down the 12 and then we'll start with the other. Step one that will be repeated. This step one is to multiply We multiply 5/3 times 12 and write it up here, 5/3 times 12 is 20. And then step two Is to add we add negative 35 plus 20 and that gives us -15. Alright, we go back to multiplying 5/3 times negative 15. That is negative 25 back to adding 28 plus a negative 25 is positive three. Back to multiplying five thirds times three equals five and adding negative five plus five is zero. And yes, this does work. There is no remainder. So five thirds is one of our roots. And now we can take this result here and make our new polynomial expression. So we've got this 12 is the coefficient for our new leading term and we're going to go one degree lower than the degree we had in the previous equation. So instead of X cubed, it's now 12 X squared and bringing down that minus 15 and then just going down one degree from the one we have here minus 15 X. And then plus three as our constant term equals zero still and 12 X squared minus 15 X plus three. Those will have a three in them. So I'm going to factor a three out in front and 12 X squared divided by three is four X squared negative 15 X divided by three is minus five X. And plus three divided by three is plus one. This is a little bit easier for me to factor in my head. So looking at four X squared what times what will give us a four X squared? Well, we've got options for that. So let's actually look over here at this plus one. What times what will give us plus one? But it's going to have to get multiplied by that for X to give us a negative five here in the middle. So let's do a minus one minus one and then what's going to get us to a negative five? Well let's keep one of these as a four X and one of them as an X. And we can always go back and quickly foil that out to check and make sure we did it correctly. First four x times X gives us four X squared Outsides four X times negative one negative four X insides negative one times X is mine minus one X and last negative one times negative one plus one, combining these inside. Like terms we have four X squared minus five X plus one. Yes we factored correctly. So we've got two factors here. We've got this four x minus one and we've got x minus one that we can both set equal to zero to solve for our other two X values. So starting on the left we'll add one to both sides and we've got four X equals one, divide both sides by four and we've got X equals 1/4. And the one on the right is a bit easier. We'll just add one to both sides and we've got X equals one. So we've got three X values floating around here, five thirds one and 1/4. We look at all of our answer choices and this matches with answer choice. A Well done. We'll catch you on the next one

Solve the equation 12x^3+16x^2−5x−3=0 given that -3/2 is a root.

Key Concepts

Polynomial Equations

Rational Root Theorem

Synthetic Division