Use synthetic division to divide f(x)=x^3−4x^2+x+6 by x+1. Use th...

Question

Use synthetic division to divide f(x)=x^3−4x^2+x+6 by x+1. Use the result to find all zeros of f.

Accepted Answer

Welcome back. I am so glad you're here. We are to use the results of synthetic division of F of X equals X cubed minus 15 X squared plus 71 X minus 105 by x minus five to determine all the zeros of F. Okay, so we are to use X minus five X minus five. That's like our x minus C. And C is what's going to go up here for our synthetic division. So let's solve for C. We can subtract X from both sides and that gives us negative five equals negative C. We'll divide both sides by negative one and that will give us C equals five. So we can plug a five up here for our synthetic division. And now we need to copy down all of the coefficients and the constant term for our synthetic division. So in front of this X cubed that's a one. And then we have negative 15 A positive 71. And this constant term -105. Now we'll just start the synthetic division, not the long division. And we recall from previous lessons that the first step in synthetic division. This step only occurs one time, is to bring down that first term will bring down this one. So it's right here. Okay? And now we'll start the step one that does repeat and that is to multiply. We are going to take this five multiply by one and write the result right here, five times one is five and step two is to add so negative 15 plus five, that is negative 10. Now we go back to multiplying five times negative 10. That's negative 50. And now adding 71 plus a negative 50 gives us 21. Now back to multiplying five times 21 that is positive 105 and adding negative plus positive 105 is zero. Great no remainder. So five is one of our zeros of F. So we can start. Well mine's a little more squiggly but one of our results is five. Now let's see what else we've got. We know from previous lessons that when we have this results of our synthetic division, that gives us a new polynomial. And this first term here is the leading coefficient for a variable that's one degree less than our original starting variable. So instead of one X to the third it's now one X squared X squared is one degree lower than that X third was from our previous expression. And then we're just going to keep going right down the line. We copy down our minus 10. That is a coefficient to a variable with a degree one lower than the one right in front of it. So X squared we're going to go down one degree two X. And then we just bring down our plus 21 as our constant term. So we have our new expression here. I'm going to erase the one in front of the X squared and we've got X squared minus 10 X plus 21 equals zero. Hey we can factor that. Let's factor at what times what gives us X squared. That's X times X. And what two numbers when multiply give us a positive 21 but when added together give us a negative 10, that's going to be a minus seven and a minus three. We now have two factors that we can set equal to zero and solve for our other two X values. So here for x minus seven equals zero, add seven to both sides and we have X equals seven. We can put seven over here in our squiggly nus and over here for x minus three equals zero will add three to both sides and we get X equals three. We can add three over here and add some squiggly nous. So our three solutions, if we were to put them in order are 35 and seven. We look at our answer choices and that matches with answer choice C. Well done, we'll catch on the next one

Use synthetic division to divide f(x)=x^3−4x^2+x+6 by x+1. Use the result to find all zeros of f.

Key Concepts

Synthetic Division

Polynomial Zeros

Remainder Theorem