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

Question

For each polynomial function, one zero is given. Find all other zeros. $ƒ (x) = 4 x^{3} + 6 x^{2} - 2 x - 1; \frac{1}{2}$

Accepted Answer

Hey, everyone in this problem, if one third is a zero of the following polynomial function we're asked to solve for all the other zeroes. The function we're given is F of X is equal to three, X cubed minus 43 X squared plus 56 X minus 14. We're given four answer choices. Option A X equals negative seven plus the square root of 35 X equals negative seven minus the square root of 35. Option B X equals seven plus the square root of X equals seven minus the square root of 35. Option C X equals 14 and X equals negative one and option D X equals negative 14 and X equals one. Now we're told that one third is a zero. OK? If one third is a zero of our polynomial, OK, then we have X equals one third and we know that X minus one third will be a factor. OK? We can think about doing the reverse order. If we have X minus one third as a factor, we set the equation equal to zero to fall to solve for the zeros. And we get that X is equal to a third. OK. So we're just working backwards here. We have our zero which tells us that we have this particular factor X minus one third. So if we have that factor, that means we can write our function F of X as that factor X minus one third multiplied by some function G of X. OK. And G of X is just whatever is left over when we factor out that term X minus the third. Now, in order to find G of X, we can take F of X and divide it by X minus one third. OK. So to find G FX, we're going to divide F of X by X minus one third. And this is the same, you can imagine if we just had numbers in this equation. OK. F FX is equal to X minus a third multiplied by G of X. If those were just numbers, in order to solve for G, we would divide by whatever X minus a third is. OK. And so we're doing the exact same thing, but now we're dividing functions. So we're gonna use long division to do this division under our division line. We're gonna include the function F of X. So we have three X Q minus 43 X squared, 56 X minus 14. OK. And on the left hand side of our division, we have what we're dividing by which is X minus one third. Now we're gonna start with the first term on the left hand side underneath our division line, we have three X cubed. And we're gonna divide that by the first term, the factor on the left hand side. OK. So we have X minus one third of that factor. The first term is X. And so we're taking three X cube divided by X, that's gonna give us three X squared and we're gonna write it up above. Now, we take that three X squared we found and we multiply it by the entire factor on the left hand side, three X squared multiplied by X minus one third. This is gonna give us the three X cubed minus X square. Now we're gonna draw another horizontal line here and we're gonna subtract the two rows we have. So we have three X cubed minus three X cube. It is a zero, we have negative 43 X squared minus negative X squared that gives us negative 42 X squared. And so we have negative 42 X squared. We bring the rest of the terms down. So we're left with plus 56 X minus and we're gonna repeat the process again. OK? But now we have this updated function to deal with. So now we're gonna take negative 42 X squared divided by X that gives us negative 42 X. We write it up above. We take that value we found negative 42 X and we multiply it by the entire factor on the left hand side, negative 42 X multiplied by X minus one third. And we're gonna write it down below. OK? Now we get negative 42 X squared S or T X. Now we're gonna subtract our two call or our two rows again. So we subtract negative 42 X squared minus negative 42 X squared gives us zero 56 X minus 14 X gives us positive 42 X and we bring our minus 14 down to this function. So now we have 42 X minus 14, we're dividing it by X minus the third. So we take the first term, 42 X divide it by the first term in our factor on the left hand side which is X 42 X divided by X gives us 42. We write it up above we multiply that value of 42 by the entire factor X minus a third. This gives us 42 X minus 14. Hey, you can see that we got the exact same function as we had left. So when we subtract the 2 42 X minus 42 X gives us zero, negative 14 minus negative 14 gives us zero and we're left with a remainder of zero, which is great. That means that this factors nicely and we can write out those factors. That's exactly what we want. Now, the value at the top, this function at the top of our division three X squared minus 42 X plus 42. Is that function G of X? OK. That's the function that's left over when we took out the factor X minus the third. And so our function F of X can be written as three X squared minus 42 X plus multiplied by X minus a third. Now we know the zero from the X minus one third factor, we need to find the zeros from this other factor. OK. So if we set our function equal to zero, in order to find the zeros, we have zero is equal to three, X squared minus 42 X plus multiplied by X minus a third. And we have these two factors multiplied together. So if either one of them is equal to zero, then this equation will be satisfied because the right hand side will be satisfied and equal to zero. OK. So if we imagine this first factor, we set it equal to 03, X squared minus 42 X plus is equal to zero. This is like solving for the roots of a quadratic function. OK. We know how to do that. We can use our quadratic formula. Now, the three value is going to be our A here, that's a coefficient associated with X squared negative 42 is going to be our B value. The coefficient associated with X and 42 is going to be our C value that constant term. And the solutions are gonna be X is equal to negative B plus or minus the square root of B squared minus four AC all divided by two. A substituting in our values, we have 42 plus or minus the square root of negative 42 squared minus four, multiplied by three, multiplied by 42. This is all divided by two multiplied by three. Now, simplifying this, we have 42 plus or minus the square root. And inside this square root symbol, if you um work out, we have 42 squared minus four multiplied by three, multiplied by 42. This is gonna give you 1260 and this is all divided by six. Now, 1260 is not a perfect square. So when we take this square, we're, we're gonna get a fraction. If you think about the answer choices, we were given, everything was left in terms of the roots, not in decimals. OK? So we don't want to turn this into a decimal. We wanna leave it as a square root. But let's try to simplify. So let's try to find a factor of 1260. That is a perfect square. And it turns out we can write this as multiplied by 35. So we have 42 plus or minus the square root of 36 multiplied by 35 all divided by six. Now 42 divided by six that gives us seven. So we have seven plus or minus the square root of 36 can be written as six. And then we have the square root of 35 left. And this part is still divided by six. Remember that divide by six applies to both terms because they're being added, those sixes will divide oh And we left when X is equal to seven plus or minus the square root of 35. OK. And so those are the other zeroes of the function we were given. Hm So going back up to the top we found and we had a zero of one third which was given in the problem. We found two additional zeros and they were X equals seven plus or minus the square root of which 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. $ƒ (x) = 4 x^{3} + 6 x^{2} - 2 x - 1; \frac{1}{2}$

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)=4x3+6x2−2x−1; 12ƒ(x)=4x^3+6x^2-2x-1;\text{ }\frac12

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. $ƒ (x) = 4 x^{3} + 6 x^{2} - 2 x - 1; \frac{1}{2}$