Solve the equation 2x3−3x2−11x+6=0 given that -2 is a zero of f(x...

Question

Solve the equation 2x3−3x2−11x+6=0 given that -2 is a zero of f(x)=2x^3−3x^2−11x+6.

Accepted Answer

Welcome back. I am so glad you're here. We are given this polynomial equation. We're told that F of X equals to X cubed plus seven, X squared minus 38 X minus 88. We're told that four is a zero of this polynomial and then we are to solve for all the rest of the zeros of this polynomial. Alright, if we know that four is zero, let's use synthetic division to then find our new polynomial and see if perhaps we can factor that new polynomial. So we're calling from previous lessons about synthetic division. We are going to copy down all of our coefficients and the constant term from our given polynomial. So in this case starting with two, we always want to start with the X degree. That's the highest. Start with its coefficient and then go down in descending order for those degrees. So after two we copy down the seven And the -38 and the -88. Alright, we've got all of those copied down. And next our first step that will only occur this one time is to bring down that first term here the two. So we write the two below the line. And now we've got a step that will be repeated. So I'm going to start a new step one and it is to multiply will take four times the two and write the result right here. Four times two is eight. The next step that will be repeated is to add we take seven plus eight, that is 15. Now we go back and do the multiplication again. Four times 15 is 60 And adding again negative 38 plus 60 is 22. Multiplying again, four times 22 is back to adding negative 88 plus 88 is zero. So yes we have no remainder and four is in fact a zero of our polynomial. We verified that. But now what we really wanted out of this is to set up our new polynomial equation. So this too is going to be the leading coefficient for a new polynomial. That's one degree less than our previous polynomial. The previous polynomial had a degree of three. So this one will have a degree of two. So it'll be two X squared. Then we just keep bringing down these terms. So that'll be plus 15 and then go down one degree from our X squared plus 15 X. And then bringing down this 22. So plus 22 And down one degree from X to the first is X to the zero. So that's just one. So 22 is our new constant term. Great. We've got our new polynomial set up. Let's see if we can factor this. So this whole thing is equal to zero and what times what will give us two X squared? Well that'll be a two X times an X. And now we have to figure out what times what will give us positive 22. But when combined With our two X and our X will add up to a positive 15. Well if we have one of them for our factors of 22 if we have one of them be a plus 11, that'll get us to 11 X. When these two multiply and then we just need to get four more. Well two x times two will give us that four more to get two plus 15 and 11 times two does give us 22. You can always check this with foil. You can set up two X plus 11 times X plus to do your first two X times X is two X squared. Outsides two X times two is plus excuse me Plus four X. Inside 11 times X is plus 11 X lasts 11 times two is plus 22 combined the middle inside like terms you get 15 X and yes this matches the equation we had before. So we factored correctly now we can set each of these factors equal to zero. So two X plus 11 equals zero and X plus two equals zero. For this first one will subtract 11 from both sides, we get two X equals negative 11, divide both sides by two. And for one of our solutions we have x equals -11/2. And for this other one we'll subtract two from both sides and we get X equals negative two. Alright, we've got three solutions. We have X equals negative two X equals negative 11 halves and X equals four. We look at our answer choices in this matches with answer choice D. Well done. We'll catch you on the next one.

Solve the equation 2x3−3x2−11x+6=0 given that -2 is a zero of f(x)=2x^3−3x^2−11x+6.

Key Concepts

Polynomial Functions

Zeros of a Polynomial

Factoring Polynomials