For each polynomial function, find all zeros and their multipl...

Question

For each polynomial function, find all zeros and their multiplicities. $ƒ (x) = (x + 1)^{2} (x - 1)^{3} (x^{2} - 10)$

Accepted Answer

Mhm Hey, everyone in this problem, we're asked to identify the zeros and their multiplicities for the following function we're given F of X is equal to X plus seven cubed multiplied by X minus seven squared, multiplied by X squared minus 45. We're given four answer choices. Option A and B have the same zeros of negative 77 and 45 but they have different multiplicities. So for a, we have a multiplicity of two for negative seven, a multiplicity of three for seven and a multiplicity of two for 45. And option B we have a multiplicity of three for negative seven. A multiplicity of two for seven and a multiplicity of two for 45. Option C and D have the zeros negative 77, the square root of 45 negative the square root of 45. The multiplicity is one for both the square root of 45 the negative the square root of 45 in both answer choices. But answer choice C has a multiplicity of three for negative seven and a multiplicity of two for seven. Whereas answer choice D has a multiplicity of two for negative seven and a multiplicity of three for seven. We're gonna start by rewriting our function F of X and F of X is equal to X plus seven cubed multiplied by X minus seven squared multiplied by X squared more minus 45. Now, this function they've given us is mostly factored. So that's helpful. So we have linear factors X plus seven and X minus seven. But we have this X squared factor at the end, we wanna go ahead and try to simplify that into linear factors. If we can, now, if we look at this, we can see that this is a difference of squares. OK. If we have a squared minus B squared, we can simplify this and factor it as A minus B multiplied by A plus B. So in this case, A is going to be our X, OK? Because we have X squared. Now we have minus B script. So B is gonna be the square root of 45 because the square root of squared gives us 45. OK. So we can factor X squared minus 45 as X minus the square root of multiplied by X about the square root of 45. And so our function F of X is X plus seven cubed multiplied by X minus seven squared multiplied by X minus the square root of 45 multiplied by X plus the squared of 45. Now, in order to solve for the zeros, we wanna set F of X equal to zero. So on the left hand side, we have zero. And on the right hand side, we have that exact same function we had before. And we aren't changing the right hand side at all. Now, we have a bunch of terms multiplied together that equals zero. If any single one of these terms is equal to zero, then the entire right hand side will be zero because they're being multiplied. OK. So we're gonna look at each of these individually if we have X plus seven cubed equal to zero, that told us that X plus seven must equal to zero. We subtract seven from both sides and we get that X must equal negative seven. Are you doing the same for the next term? We have X minus seven squared equal to zero. That means that X minus seven must equal to zero. And so X must equal seven. Our next term we have X minus the square root of equal to zero, which tells us that X is equal to the square root of 45 K adding that square root of 45 to both sides. And finally, we have X plus the square root of 45. We set that equal to zero. We subtract the square root of 45 from both sides. And we get that X is equal to negative the square root of 45. And so we've found our zeros, we have X equals negative seven, X equals seven X equals the square root of 45 X equals negative the squared of 45. So we know that we're looking at either answer C or D and now we need to look at the multiplicity. Now, when we say multiplicity of a zero, what we mean is how many times that zero appears? OK. So when we look at our function, we want to look at how many times the factor associated with a particular zero appears in our equation. So for X equals negative seven, we have the factor X plus seven and that has an exponent of three. And so for X equals negative seven, this is gonna have a multiplicity of three. We can do the same for X equals seven. OK. The factor associated with X equals seven is X minus seven and it has an exponent of chip. OK. It's squared and so X equals seven has a multiplicity of chip. Now, for X equals square root of 45 X equals negative to square root of 45 they both have exponents of one. And so they both have a multiplicity of just one. And so we've solved the entire problem, we found that we have the root negative seven with a multiplicity of three, the root seven with a multiplicity of two and the roots um square root of 45 negative the square root of 45 with multiplicities of one. So this corresponds with answer choice. C thanks everyone for watching. I hope this video helped see you in the next one.

For each polynomial function, find all zeros and their multiplicities. $ƒ (x) = (x + 1)^{2} (x - 1)^{3} (x^{2} - 10)$

Key Concepts

Polynomial Zeros

Multiplicity of Zeros

Factoring and Solving Quadratic Expressions

For each polynomial function, find all zeros and their multiplicities. ƒ(x)=(x+1)2(x−1)3(x2−10)ƒ(x)=(x+1)^2(x-1)^3(x^2-10)

Key Concepts

Polynomial Zeros

Multiplicity of Zeros

Factoring and Solving Quadratic Expressions

For each polynomial function, find all zeros and their multiplicities. $ƒ (x) = (x + 1)^{2} (x - 1)^{3} (x^{2} - 10)$