Finding Roots of Higher Order Polynomials

Quadratics

 Use factoring when the roots are rational f(x)=x 2 – 5x – 14  Use the quadratic formula if it is not factorable or if you’re not sure

x

 

b



b

2

a

2  4

ac

Roots of Cubic Functions

     Find the roots (zeros) of

f

(

x

) 

x

3  6

x

2  11

x

 6 First, use the table feature of your calculator.

Second, inspect the graph of the function.

Third, choose one of the values you believe to be a zero and use long division by the factor associated with it. If k is a zero, then x – k is a factor. The quotient will be a quadratic, so use previous methods to find the remaining zeros.

Synthetic Division

    This is a less cumbersome way to accomplish the same thing as long division.

You must divide by a zero associated with a factor.

Since you begin with the coefficient associated with x

n

, you must have a coefficient for every power of x from n to zero.

A remainder of zero indicates that the number you divided by is a zero.

Rational Roots Theorem

     How do you know what to divide by if you can’t use a graphing calculator?

Every rational zero of a function, f, has the form rational zero =

p q

p is a factor of the constant term of the function q is a factor of the leading coefficient of the function What are the possible rational zeros of

f

(

x

) 

x

3  6

x

2  11

x

 6 ?

Remainders

 How do you know which of the possible rational zeros work?

 If dividing by k gives a remainder of zero, then  k is a zero of the function  (x – k) is a factor of the function  The point (k , 0) is an x-intercept of the function.

 If dividing by k does not give a remainder of zero, the remainder is the value of the function at k. That is f(k)=remainder

Descartes’ Rule of Signs

   Descartes' Rule of Signs will not tell you where the polynomial's zeroes are (you'll need to use the Rational Roots Test and synthetic division, or draw a graph, to actually find the roots), but the Rule will tell you how many roots you can expect.

First, I look at the polynomial as it stands, not changing the sign on x, so this is the "positive" case: Now I look at f (–x) (that is, having changed the sign on x, so this is the "negative" case): For detailed information go to: http://www.purplemath.com

Let’s Practice

f

(

x

)  6

x

3  19

x

2  16

x

 4

f

(

x

)  2

x

3  11

x

2  9

x

 2

f

(

x

) 

x

3  7

x

2  17

x

 14

f

(

x

)  2

x

4  7

x

3  4

x

2  27

x

 18