2. Polynomial Functions
  3. Equations of Polynomial Functions in Factored Form

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Families of Polynomial Functions

Families of Polynomial Functions Example 1

Families of Polynomial Functions Example 2

Families of Polynomial Functions Example 2

Determine the general equation of a quartic function with end behaviour \(f(x) \rightarrow -\infty\) as \(x \rightarrow \pm\infty\), a zero at \(x=\frac{1}{2}\), and another at \(x=4\) of multiplicity \(3\).

Solution

 

Check Your Understanding A

State the equation of the general polynomial function with the following zeros:

(r)*1.0 (((o)*(r))*(d))*1.0 (c)*1.0 (r)*2.0 (((o)*(r))*(d))*2.0 (c)*2.0 (r)*3.0 (((o)*(r))*(d))*3.0 (c)*3.0 (r)*4.0 (((o)*(r))*(d))*4.0 (c)*4.0 (r)*5.0 (((o)*(r))*(d))*5.0

Use 'k' as the general coefficient for the family. 

For example, enter f(x) = k x(x - 1)^2(x + 4)^3 for  f(x) = k x(x - 1)2 (x + 4)3

 

f(x) =  There appears to be a syntax error in the question bank involving the question field of this question. The following error message may help correct the problem:

null

Preview Change entry mode 

(Click on Equation Editor Symbol to access the Equation Editor.)

​

We can use the zeros given to create the factors in the general equation. The order of the zero provides the number of occurrences of that factor in the equation, thus providing the exponent of the factor. The general equation has a coefficient "k" to create the equation for the family of polynomial functions with the zero properties stated.

The general form of these equations is given by

fx=k⋅x−r1n1x−r2n2x−r3n3...

where r1, r2, ... are the zeros and  n1, n2,... are the orders of the zeros. Using this general form, the function is: f(x) = ((((A)*(n))*(s))*(M))*(L)

 

 

Check Your Understanding B

State the equation of the general polynomial function with the following zeros:

(((r)*1.0)*(r))*(a) (((r)*1.0)*(r))*(b) ((((o)*(r))*(d))*1.0)*(A)((c)*1.0)*(A) ((r)*2.0)*(A) ((((o)*(r))*(d))*2.0)*(A)((c)*2.0)*(A) ((r)*3.0)*(A) ((((o)*(r))*(d))*3.0)*(A)((c)*3.0)*(A) ((r)*4.0)*(A) ((((o)*(r))*(d))*4.0)*(A)((c)*4.0)*(A) ((r)*5.0)*(A) ((((o)*(r))*(d))*5.0)*(A)

Use 'k' as the general coefficient for the family. 

For example enter  f (x) = k x (2x - 3)^2 (x - 2)^3 for f (x) = k x (2x - 3)2 (x - 2)3.

 

f(x) = There appears to be a syntax error in the question bank involving the question field of this question. The following error message may help correct the problem:

null

Preview Change entry mode 

 

​

We can use the zeros given to create the linear factors in the general equation. The order of the zero provides the number of occurrences of that factor in the equation, thus providing the exponent of the factor. The general equation has a coefficient "k" to create the equation for the family of polynomial functions with the zero properties stated.

The general form of these equations is given by

fx=k⋅x−r1n1x−r2n2x−r3n3...

where r1, r2, etc. are the zeros and n1, n2, etc. are the order of the corresponding zeros.

Following this equation, we know the final function to be:

fx=k⋅((b)*11.0)*(A)((((f)*(r))*1.0)*(r))*(b)((x)*1.0)*(A)(((p)*(m))*1.0)*(A)((((f)*(r))*1.0)*(r))*(a)((b)*12.0)*(A)((((f)*(o))*(r))*1.0)*(A)((b)*21.0)*(A)((x)*2.0)*(A)(((p)*(m))*2.0)*(A)((((f)*(r))*2.0)*(r))*(A)((b)*22.0)*(A)((((f)*(o))*(r))*2.0)*(A)((b)*31.0)*(A)((x)*3.0)*(A)(((p)*(m))*3.0)*(A)((((f)*(r))*3.0)*(r))*(A)((b)*32.0)*(A)((((f)*(o))*(r))*3.0)*(A)((b)*41.0)*(A)((x)*4.0)*(A)(((p)*(m))*4.0)*(A)((((f)*(r))*4.0)*(r))*(A)((b)*42.0)*(A)((((f)*(o))*(r))*4.0)*(A)((b)*51.0)*(A)((x)*5.0)*(A)(((p)*(m))*5.0)*(A)((((f)*(r))*5.0)*(r))*(A)((b)*52.0)*(A)((((f)*(o))*(r))*5.0)*(A)


 

 

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