Form A Polynomial Function

[mathdad]

Given zeros -2, 2, 3 and degree 3, form a polynomial function.

NOTE: I seek solution steps only. I want to try this on my own (following your steps) before posting my work here.

 

[MarkFL]

The family of polynomials of degree \(n\) having the roots \(r_j\) where \(1\le j\le n\) is given by:

[MATH]f(x)=k\prod_{j=1}^n(x-r_j)[/MATH] where \(k\ne0\)

 

[mathdad]

The family of polynomials of degree \(n\) having the roots \(r_j\) where \(1\le j\le n\) is given by:

[MATH]f(x)=k\prod_{j=1}^n(x-r_j)[/MATH] where \(k\ne0\)

I do not get it.

 

[Dr.Peterson]

Given zeros -2, 2, 3 and degree 3, form a polynomial function.

I suppose you didn't get MarkFL's answer because you don't know product (Pi) notation. That's not too surprising.

Your book presumably says the same thing in a different way, perhaps in the form of something called the "factor theorem". I'll explain it by way of example:

Consider first the polynomial (in factored form) f(x) = 2(x - 4)(x + 1). Its zeros are 4 and -1, the numbers that make a factor zero.

Now reverse that. Suppose you want a function whose zeros are 4 and -1. You can make factors (x - 4) and (x + 1) that will be zero when x has the two given values - that is, (x - r) for each zero r. Then the function f(x) = a(x - 4)(x + 1) will have those zeros, regardless of what value you put in for a (except 0, of course). The simplest answer to give would be f(x) = (x - 4)(x + 1), using a=1.

Now do the same for your problem.

 

[Otis]

I do not get it.

Maybe these three pages will help you with defining polynomials.


https://www.mathsisfun.com/algebra/polynomials.html

\(\;\)

 

[Otis]

Given zeros -2, 2, 3 and degree 3, form a polynomial function.

NOTE: I seek solution steps only.

Step 1: Use each zero to write a factor of the polynomial

EG: If number r is a zero, then (x-r) is a factor

Step 2: Multiply the factors together

?

 

[mathdad]

I suppose you didn't get MarkFL's answer because you don't know product (Pi) notation. That's not too surprising.

Your book presumably says the same thing in a different way, perhaps in the form of something called the "factor theorem". I'll explain it by way of example:

Consider first the polynomial (in factored form) f(x) = 2(x - 4)(x + 1). Its zeros are 4 and -1, the numbers that make a factor zero.

Now reverse that. Suppose you want a function whose zeros are 4 and -1. You can make factors (x - 4) and (x + 1) that will be zero when x has the two given values - that is, (x - r) for each zero r. Then the function f(x) = a(x - 4)(x + 1) will have those zeros, regardless of what value you put in for a (except 0, of course). The simplest answer to give would be f(x) = (x - 4)(x + 1), using a=1.

Now do the same for your problem.

Perfect and easily explained.

 

[mathdad]

Maybe these three pages will help you with defining polynomials.


https://www.mathsisfun.com/algebra/polynomials.html

\(\;\)

Great links. Thank you.