Find a polynomial f that satisfies the following properties. (Hint: Determine the degree of f; t

Find a polynomial f that satisfies the following properties....

Key Concepts

Substitution Method

The substitution method involves assuming a general form for the polynomial (with undetermined coefficients) based on its expected degree and then substituting this form into the composition. This step translates the problem of finding a specific polynomial into finding the correct coefficients that match the target expression.

Equating Coefficients

After composing the polynomial with itself and expanding the expression, the equating coefficients technique is used to set the coefficients of like powers of x equal to those in the target polynomial. This generates a system of equations that, once solved, provides the values of the unknown coefficients and hence the polynomial f.

Polynomial Composition

Polynomial composition involves applying one polynomial function to the output of another. In the context of the problem, the operation f(f(x)) requires substituting the polynomial f(x) into itself, which necessitates careful consideration of how the degrees and coefficients interact to produce the resulting polynomial.

Degree Analysis

Degree analysis is used to determine the appropriate degree of the polynomial f. Since composing polynomials multiplies their degrees, analyzing the degree of the target polynomial helps infer the degree of f. In problems like this, if f(f(x)) has a known degree, it suggests a specific degree for f, making it possible to set up a general form.

Transcript

00:01 Hello, everybody.

00:03 In this video, i'm going to be showing you how to solve exercise 95 in chapter 1, section 1 of calculus early transcendentals.

00:12 Now, this problem asks us to find a polynomial f so that f of f of x is equal to x of the 4 minus 12x squared plus 30.

00:23 To do this, let's first acknowledge the degree of the output of this expression, which is 4.

00:30 Now, if we plug f of a certain degree into itself, then of course the output is going to entail raising to the power x to the degree of that same power.

00:44 So therefore, we can see that this expression is going to have a degree that is the square of the degree of x.

00:51 All in all, this means that f is going to have degree two.

01:00 And therefore, we can say that f takes the general form, f of x, equals a x squared plus bx squared i'm sorry just bx plus c and so this problem reduces to just figuring out what these three constants ab and c are to do this let's plug f into itself we have the f of f of x is equal to f of a x squared plus bx plus c and this is where it gets a little bit messy we have that this equal to a times a x squared plus bx plus c squared plus b times a x squared plus b times a x squared plus bx let's see and then c is added to all of this so again this gets really messy here and unfortunately all we can really do is kind of foil this term out and distribute this quantity so let's just go ahead and do all that we have that this point that we previously got.

02:35 If we foil the squared term, then we get a times a squared x to the fourth plus ab x cubed plus ac x squared plus ab x squared plus ab x squared plus ab x squared plus ab x squared plus b, c, x, plus c squared.

03:27 Maybe let's get this out here so we can see it better, plus c squared.

03:37 And now we distribute that second term, and what we get is that this is equal to a, b, x squared, plus bx, b squared x, plus b c.

03:54 And then we have c added to this.

04:00 So the best way of organizing all these terms now is to organize it by a degree.

04:05 Degrees of x.

04:08 Now grouping everything by the degree of x is really a trivial exercise, so i'm going to leave you to figure that out, just how to look at every term on its own and figure out where everything goes.

04:20 But with that, i'm just going to write out what this becomes.

04:25 We get that this is equal to a cubed x to the fourth plus to a squared b x cubed x cubed plus to 2, a squared, c plus a, b squared, plus ab squared, plus ab, x squared...

Please give Ace some feedback