For each polynomial function, one zero is given. Find all othe...

Question

For each polynomial function, one zero is given. Find all other zeros. $ƒ (x) = x^{3} - x^{2} - 4 x - 6; 3$

Accepted Answer

Hey, everyone. This problem asks if five is a zero of the following polynomial function solve for all the other zeros. The function we're given is F of X equals X cubed minus nine, X squared plus 25 X minus 25. Option A has X is equal to negative two plus or minus. I has two solutions. Option B also has two solutions. X is equal to two plus or minus. I option C has XX is equal to negative one and X is equal to five. And option D has X is equal to one and X is equal to negative five. Now, we're told that five is a zero of this polynomial. If five is a zero of this polynomial, that means that X minus five is a factor. We know that when we're solving for zeros, if we have our polynomial factored, we set it equal to zero. OK, define those zeros. And then we can set each factor individually to zero. So if we have the factor X minus five, we set this equal to zero, we get that X is equal to five and so five is a zero. OK? So we're just working backwards here, we're starting with the zero, we're writing the factor. So if X minus five is a factor, that means our function F of X can be written as X minus five multiplied by some function G of X and G of X is gonna be whatever is left over when we factor out X minus five. Now, in order to figure out what G of X is, we need to divide by X minus five. OK. So we're gonna take F of X and divide it by X minus five in order to determine G because the roots of G are gonna be the remaining roots of this polynomial. So the fine G we're gonna divide of by X minus five. Now we're gonna do this using long division. So we write out our long division um table or line underneath the line. We write our function X cubed minus nine, X squared plus 25 X minus 25. On the left hand side, we write the factor we're dividing by X minus five. We're gonna start on the left hand side underneath our line. We have X cubed. We divide that by the first term in our factor. On the left hand side, X cube divided by X gives us X squared. We write it above, then we take X squared and we multiply it by the entire factor X minus five, X squared, multiplied by X minus five, gives us X cubed minus five. X squared. We're gonna write that below our original function and we're gonna keep the similar exponents in the same columns. OK. So the X cube terms stay in a column, the X squared terms stay in a column, et cetera. And now we're gonna subtract these two rows. We have, we have X cubed minus X cubed which gives us zero, we have negative nine X squared minus negative five X squared which gives us negative four X squared. The remaining terms of our original function we're gonna bring down, there was nothing to subtract there. So we still have plus 25 X minus 25. And we have this new function negative four X squared plus X minus 25. We're gonna repeat the same process again. With this new function, we take negative four X squared, we divide it by X that gives us negative four X. We write it above, we multiply that negative for X by the entire factor X minus five. This gives us negative four X squared less X. We write it down below keeping the similar exponents in the same column. And now we subtract negative four X squared minus negative four X squared gives us zero 25 X minus 20 X gives us five X and we bring down our constant. We still have minus 25 in our equation five X minus 25 we have this new function, we repeat the process one more time five X multi or sorry, divided by X gives us five. So we have plus five, we write it up above. We multiply that five by the entire factor X minus five, we get five X minus 25 we write it down below and you'll notice that's the exact same function. So when we subtract, we have five X minus 25 minus five X minus 25 which gives us zero. So we have this remainder zero, which is what we want. The function at the top of our division is the value of G of X we were looking for, OK. So we have that G of X is X squared minus four, X plus five. And so our function at the X can be written as X minus five multiplied by X squared minus four X plus five. Now, what we wanted was to find these other zeros. So we want to set this function equal to zero and solve for the zeros. We get zero is equal to X minus five multiplied by X squared minus four, X plus five. We know the first factor X minus five gives us a zero of X equals five. So let's look at when this second factor is equal to zero. If this second factor is equal to zero, the entire right hand side will be equal to zero. We have zero equals zero. So that satisfies the equation. So we wanna look at when X squared minus four, X plus five is equal to zero. Now, this is a quadratic equation equal to zero. We can use our quadratic formula. OK. Now A is going to be the coefficient corresponding to the X squared term, which is just one. In this case, B is gonna be the coefficient corresponding to our X term which is negative four and C is going to be our constant term of buy. And so our quadratic formula recall is X is equal to negative B plus or minus the square root of B squared minus four ac divided by two. A substituting in the values we have, we get that B is equal to four plus or minus the square root A negative four squared minus four multiplied by one multiplied by five. And all of this is divided by two multiplied by one. Now we can simplify this. We get that this is equal to four plus or minus. Now, under our square root, we have negative four squared which gives us 16 minus four multiplied by one multiplied by five. So that's minus 2016 minus 20 gives us negative four. And so we have four plus or minus the square root of negative four divided by two. Now negative four, we can write four multiplied by negative one. So we get four plus or minus the square root of four multiplied by negative one, all divided by two. This is equal to four plus or minus. Now the square root of four gives us two. So we get two multiplied by the square root of negative one, all divided by two. And recall that the squared of negative one is equal to I. So we have four plus or minus two, I all divided by two, which gives us two plus or minus I. And so the roots of this function G of X that we found X squared minus four, X plus five are gonna be X equals two plus or minus I. And these are the two extra roots from our original function F FX K F FX was a cubic function. It has three roots. One of them was X equals five which we were given. And we've found that the other two are X equals two plus or minus. I, we look at our answer choices. We can see that this corresponds with answer choice. B thanks everyone for watching. I hope this video helped see you in the next one.

For each polynomial function, one zero is given. Find all other zeros. $ƒ (x) = x^{3} - x^{2} - 4 x - 6; 3$

Key Concepts

Polynomial Zeros and Roots

Polynomial Division (Synthetic or Long Division)

Factoring and Solving Quadratic Equations

For each polynomial function, one zero is given. Find all other zeros. ƒ(x)=x3−x2−4x−6; 3ƒ(x)=x^3-x^2-4x-6;\text{ }3

Key Concepts

Polynomial Zeros and Roots

Polynomial Division (Synthetic or Long Division)

Factoring and Solving Quadratic Equations

For each polynomial function, one zero is given. Find all other zeros. $ƒ (x) = x^{3} - x^{2} - 4 x - 6; 3$