Solve each problem. Find a polynomial function ƒ of degree 3 with...

Question

Solve each problem. Find a polynomial function ƒ of degree 3 with -2, 1, and 4 as zeros, and ƒ(2)=16.

Accepted Answer

Identify the polynomial function given the following features. Degree is three. Our zeros are negative 52 and six and F of four equals 72. Now, to solve this, let's first find our factors. So it tells us the zeros are negative 52 and six and our degree is three. That means we will have three factors for a polynomial. We have X equals negative five, which means one of our zeros will be X plus five as factor, we also have X minus two and X minus six. This will be Y equals all three of these factors multiplied together all multiplied by A to account for a potential constant multiplier in the front. Let's simplify the multiplication of our three factors. We would first ski X square minus two, X plus five, X minus 10, multiplied by X minus six, giving us X squared plus three, X minus 10, multiplied by X minus six. We can now multiply our next set of polynomials X squared times X gives us X to the third X squared multiplied by negative six, gives us negative six, X squared three, X and X gives us positive three, X square three X and negative six gives us negative 18, X, negative 10 multiplied by X gives us negative 10, X and negative 10 multiplied by negative six gives us positive 60. Meaning we have X the third minus three, X squared minus 28 X A 60 which is all multiplied by our constant multiplier. A set equal to Y. Now we can solve for R A by using our F of four equals 72. So we have 72 equals a multiplied by four to the third minus three, four to the second minus 28 multiplied by four plus can now simplify 72 equals A P P by minus 48 minus 1, 12 plus 60. We combine everything inside the parentheses. We get 72 equals a multiplied by negative 36 dividing both sides by negative 36 gives us a equals negative two. Our final function as in Y equals multiplied by X to the third minus three X squared minus 28 X plus 60. And we can simplify once again by just multiplying the negative two to get negative two X to the third plus six X squared plus 56 X minus 20. We then notice he answers to our problem corresponds to answer B OK. I hope to help you solve the problem. Thank you for watching. Goodbye.

Solve each problem. Find a polynomial function ƒ of degree 3 with -2, 1, and 4 as zeros, and ƒ(2)=16.

Key Concepts

Polynomial Functions

Zeros of a Polynomial

Evaluating Polynomial Functions