{Use of Tech} Polynomial calculations
Find a polynomial&...

Question

{Use of Tech} Polynomial calculations

Find a polynomial ƒ that satisfies the following properties. (Hint: Determine the degree of ƒ; then substitute a polynomial of that degree and solve for its coefficients.)

ƒ(ƒ(x)) = x⁴ - 12x² + 30

Accepted Answer

Hello. Today, we're going to be identifying the function u of x for the given composite function. So the given composite function is u of u of x, and this composite function is equal to x raised to the power of 4 minuteus 6x squared plus 6. Now one thing to note is that the composite function given to us is in the form of a quadratic function. We can allow u of x to equal to a generic quadratic function and state that u of x is equal to x squared plus a. Now the reason why we state that u of x has a leading coefficient of 2 or a leading exponent of 2 is because in the given composite function, the leading exponent is 4, which is an even exponent. U of x cannot have an odd numbered leading exponent. Otherwise, we would have an odd numbered leading exponent in the given composite function. So in order to identify the function u of x, we're going to need to solve for the a variable in our function. If we were to plug in the generic form of u of x into itself, we can state that the composite function u of u of x is equal to x squared plus a, that quantity squared, plus a. Now if we were to foil the right hand side of this comp or this composite function, we can rewrite the right hand side as x raised to the 4th power plus 2 a x squared plus a squared plus a. Now on the left hand side, we are stating that the composite function u of u of x is equal to this generic composite function. But we do know that the final form of the composite function should equal to x raised to the power of 4 minuteus 6x2+6. So we can plug in this form of the composite function to the left hand side of the equation and state that u of u of x is equal to x raised to the power of 4 minuteus 6x squared plus 6. Now in order to solve for a, we can take a look at the following coefficients. Notice that on both sides of the equation, we have an x squared term and another x squared term. We can set both of the leading coefficients of the x squared terms equal to each other in order to solve for a. Doing so will allow us to state that negative 6 is equal to 2 a. And in order to solve for a, we'll just have to divide both sides of this equation by 2 and that will leave us with a is equal to negative 3. So now that we have the value for a, we can plug this value of a into the general form of u of x and that will allow us to say that u of x is equal to x squared minus 3. This is going to be the function u of x that we are looking for. And with that being said, the answer to this problem is going to be b. So I hope this video helps you in understanding how to approach this problem, and I'll go ahead and see you all in the next video.

{Use of Tech} Polynomial calculations
Find a polynomial ƒ that satisfies the following properties. (Hint: Determine the degree of ƒ; then substitute a polynomial of that degree and solve for its coefficients.)
ƒ(ƒ(x)) = x⁴ - 12x² + 30

Key Concepts

Polynomial Functions

Composition of Functions

Degree of a Polynomial

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