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

Question

For each polynomial function, one zero is given. Find all other zeros. $ƒ (x) = x^{3} + 4 x^{2} - 5; 1$

Accepted Answer

Hey, everyone. This problem asks if one 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 plus 13, X squared minus 40. We're given four answer choices. Option A X is equal to negative seven plus the square root of 35 X is equal to negative seven minus the square root of 35. Option B X is equal to seven plus the square root of X is equal to seven minus the square root of 35. Option C X is equal to 14 and X is equal to negative one or option D X is equal to negative 14 and X is equal to one. Now, we're told that one is a zero of this polynomial. If one is a zero of this polynomial, the top of the X minus one is a factor of this polynomial. Now, why is that? We can think about this backwards. If X minus one is a factor, we set our polynomial equal to zero to solve for the zeros. So we can set our factor equal to zero to solve. For one of the zeros, we have X minus one equals zero, which tells us that X is equal to one. So one is the zero. OK? So we're just doing the reverse of what we typically do when we solve for zeros. OK. We're starting with the zero. We're writing at the factor. Now, if X minus one is a factor, that means that our function F of X can be written as that factor X minus one multiplied by G of X. Some function in that function G of X is gonna be whatever is left over after we factor out X minus one. Now the rest of the zeroes of this polynomial are gonna come from G of X. OK. So how can we find geo well, if we divide our function F of X by this factor X minus one, we'll get G of X. OK. It's just like solving this equation for G of X G of X is equal to F of X divided by X minus one. OK. In this case, we're dividing functions instead of just dividing numbers. Now, in order to do this division, we're going to use long division. So we draw our long division line and underneath it, we're gonna write the function F of X that's cued last 13 X word. Now, in our function, we aren't given the X term. OK? We have no X term, but we want to include it in our long division. It's important to include all of the exponents from the highest exponent down to the constant term. And so we're gonna write it as plus zero X and then we have minus 14 at the end on the left hand side, we're gonna write the factor we're dividing by which is X minus one. Now to perform the division, we're gonna take the first term on the left hand side underneath our division sign which is X cubed. And we want to divide it by the first term in our um factor on the left hand side. OK. So we're gonna take X cubed divided by X. This gives us X squared and we write it above, we take that value, we found X squared and multiply it by the entire factor X minus one on the left. So X squared multiplied by X minus one gives us X cubed minus X squared. And we write this down below underneath our original function so that the exponents stay in the appropriate columns. OK? So we have all the X cubes in the same column. We have the X squared in the same column. And now we're gonna subtract those two rows. We have X cube minus X Q which gives us zero in the first column. We have 13 X squared minus negative X squared in the second column, which gives us 14 X squared. And our other values can be brought down because there's nothing to subtract. So we have the plus zero X minus 14. So now we have this new function 14 X squared plus zero X minus 14 that we want to deal with. We're gonna do the same thing. We're gonna take the first value on the left hand side, 14 X squared divided by the first term on the left-hand side, X 14 X squared divided by X gives us X. So we write our plus 14 X above. Now we take that value 14 X, we multiply it by the entire factor X minus one, we get 14 X squared minus X. Let me write it down below again. In corresponding column, the X squared terms are in one column. The X terms are in another and we subtract 14 X squared minus 14 X squared gives us zero zero X minus negative 14 X gives us positive 14 X and we bring our negative 14 down because there's nothing to subtract there. So it stays as it is. Now we have a new function 14 X minus 14. We repeat the process again. 14 X divided by X gives us positive 14. We rate it above 14, multiplied by X minus one gives us 14, X minus 14. We write it at the bottom in the corresponding columns and we subtract 14, X minus 14 X gives us zero, negative 14 minus negative 14 gives us zero. And so we get a remainder of zero, which is what we wanted. So now we have what we want. The top function that we found X squared plus 14 X plus 14. That's the result of the division. And that's the function G FX we were looking for. And so now we can write our function F of X as X minus one multiplied by X squared plus 14, X plus 14. So remember what this question wanted us to do. It wanted us to find all of the other zeroes. OK. So to find the zeros, we set our function equal to zero, we have zero is equal to X minus one, multiplied by X squared plus 14, X plus 14. We know that X minus one gives us a zero X equals one. We started with that in our problem. So what we're gonna do now is set this second factor equal to zero. If this second factor is equal to zero, the entire right hand side is equal to zero. And so we get a solution. And so we have X squared plus 14, X plus 14 equal to zero. This is a regular quadratic function equal to zero. We can use our quadratic formula. Now recall our A is going to be equal to one. That's the coefficient corresponding with the X squared term B is going to be 14, the coefficient corresponding with our X term and C is going to be 14, that constant term. Our quadratic formula says that X is equal to negative B plus or minus the square root of B squared minus four AC all over two. A substituting in the values we have, we get that X is equal to negative 14 less or minus the square root of squared minus four multiplied by one multiply by 14, all divided by two, multiplied by one. We can simplify this. We get negative 14 plus or minus the square root of 140 divided by two. Now 140 is not a perfect square. When we take the square root, we will get a decimal. We're not gonna get a nice whole number. The answer choices we were given leave the potential answers in terms of the square roots. So we wanna leave this as a square root, but we want to simplify as much as we can. So we wanna look for factors of 140 that are perfect squares and it turns out that four is one of them. So we can write this as negative 14, less or minus the square root of four multiplied by 35 divided by two. Now negative 14 divided by two gives us negative seven plus or minus. The square root of four is equal to two. So we can write this two multiplied by the square root of divided by two. OK? Don't forget that we still have the divided by two. In this second term in the numerator, these things are added and subtracted. And so the division by two has to apply to each of them. And so we get our X values are equal to negative seven plus or minus the square root of 35. And so those are the two roots of X squared plus 14, X plus 14, which are the other roots of the original function F of X we were given, OK. And so if we go back to the top, we had a cubic polynomial three roots, we were given the root X equals one. And we found that there were two additional routes negative seven plus or minus the square root of 35 which corresponds with answer choice. A 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. $ƒ (x) = x^{3} + 4 x^{2} - 5; 1$

Key Concepts

Polynomial Zeros and Roots

Polynomial Division (Synthetic or Long Division)

Factoring Quadratic Polynomials

For each polynomial function, one zero is given. Find all other zeros. ƒ(x)=x3+4x2−5; 1ƒ(x)=x^3+4x^2-5;\text{ }1

Key Concepts

Polynomial Zeros and Roots

Polynomial Division (Synthetic or Long Division)

Factoring Quadratic Polynomials

For each polynomial function, one zero is given. Find all other zeros. $ƒ (x) = x^{3} + 4 x^{2} - 5; 1$