Find constants b and c in the polynomial p(x)=x^2+bx+c such th...

Question

Find constants b and c in the polynomial p(x)=x^2+bx+c such that lim x→2 p(x) / x−2=6. Are the constants unique?

Accepted Answer

Below there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine the constraints A and D in the polynomial Q X equals X2 + A X plus D such that the limit as X approaches 3 of Q X divided by X minus 3 is equal to 7. Awesome. So it appears for this particular problem, we're trying to solve for two separate answers. We're trying to determine the values of the constraints A and D in this specific polynomial function. Awesome. So now that we know that we're solving for the values of A and D, let's read off our multiple choice answers to see what our final answer pair might be, noting we're going to read off our answer for A first and then our answer for D. So A is 1 and 12, B is -1 and 12, C is 1 and 12, and D is -1 and 12. OK. So first off, what we need to do is we need to define our known variables and the target variables, which the target variable or variables in this case is what we're ultimately trying to solve for. So we're told what we know is that we know that Q X, we know that Q of X is equal to X2 + A X plus D, and we also know that we are given the limit as X approaches 3 of Q X divided by X minus 3, and that this is equal to 7. And in blue, let's write down our target variables. We're trying to solve for the constant, so we're trying to solve for our constant. Values of A and D. Awesome. So now that we know that we're trying to solve for our constants A and D, our next step is we need to recall and use the definition of the limit and polynomial division. So we need to recall a note that sends the limit as X approaches 3. So when you say that the limit as X approaches 3, Of Q of X, so you take Q of X divided by X minus 3. is equal to 7. We need to say that this must be divisible by, and let's write this in red, so this has to be divisible by X minus 3. So with the remainder of 0, when X equals 3. So, therefore, as a result, because we need to take this specific limit, and this limit has to be divisible by X minus 3. With a remainder of 0 when X equals 3, so therefore, as a result, Q of 3 must be equal to 0. So now, our next step is we need to substitute in the value of X equals 3 into our equation, or rather our function of Q of X. So when we do that, we will find that Q of 3. is equal to 32 + 3 multiplied by A plus D is all equal to 0. So now we need to simplify this expression or this equation, and when we do, we will find that it's equal to 9 + 3 multiplied by A. Plus D. And then this will be all equal to 0, or we can go ahead and write that 3 multiplied by A. Plus D. is all equal to -9. Awesome. So now, our next step is we need to recall and use the limit condition to help us find the second equation. And as we should recall, the limit condition implies that the derivative of Q of X at X equals 3 will equal 7. So, we need to now simplify to get the second equation, so we need to take 6 plus A equals 7, or A equals 1 when we rearrange this expression to isolate and solve for A, and when we do, we'll find that A is equal to 1. Fantastic. So now our next step is we need to solve for the system of equations, so we need to substitute A equals 1 into our first equation, so meaning we need to take 3, so 3 multiplied by 1, so we need to take 3 multiplied by 1. And we need to add D, so 3 multiplied by 1, plus D is equal to -9, and once again we're substituting in the value of A into our expression. So now we need to simplify our expression to solve for the value of D. So we use a little bit of algebra, we will find that D is equal to -12. Awesome, and that's once again when we're rearranging our 3 multiplied by 1 plus D equals -9 expression, and we're rearranging it to isolate and solve for D. Awesome. So now our final step is we need to recall our methodology to help us verify the uniqueness of the constraints. So as we should recall and note, the system of linear equations, which we'll write this in red, so this is 3 multiplied by A plus D is equal to -9 and A equals 1 as a unique solution. So therefore, the constraints A and D are unique. So A and D equals unique, and I'll just do a capital U with quotation marks. So this means unique. So A and D are constraints A and D are unique. Awesome. So that means without a doubt, our final two answers are A equals 1 and D equals negative 12. And this corresponds with multiple choice sensor letter A, which once again is A equals 1 and D equals -12. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Find constants b and c in the polynomial p(x)=x^2+bx+c such that lim x→2 p(x) / x−2=6. Are the constants unique?

Key Concepts

Limits

Polynomial Functions

L'Hôpital's Rule

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