Finding all real zeros

of a PolynomialAnother example….

Find all the real zeros of

2,1632,1

6432)( 23 xxxxf

Use the Rational Zeros Theorem to make a list of possible rational zeros

23,

21,6,3,2,1

Find all the real zeros of

Use your graphing calculator to narrow down the possible rational zeros

the function seems to cross the x axis at these points…..

we’ll use the remainder/factor theorem to be sure….

6432)( 23 xxxxf

?23

x

?23

x

Find all the real zeros of

6234

233

232)

23(

23

f

Use the remainder and factor theorems to test the possible zeros

0

5.1

since the remainder is zero, (x – 3/2) is a factor!

since the remainder is not zero, (x + 3/2) is not a factor

6432)( 23 xxxxf

6234

233

232)

23(

23

f

Find all the real zeros of

Use the divisor to divide the dividend

3/2 2 -3 -4 6

2 0 -4 0

3 0 -6

So the dividend is equal to:

)42)(23( 2 xx

6432)( 23 xxxxf

Find all the real zeros of

Synthetic division has allowed us to begin factoring the polynomial, now we can use other factor techniques to take care of the rest!

Factor out the GCF

And then use difference of two squares method to factor one last time

)2)(23(2 2 xx

)42)(23( 2 xx

)2)(2)(23(2 xxx

6432)( 23 xxxxf

Find all the real zeros of 6432)( 23 xxxxf

Now that you have the polynomial in factored form, find those zeros!!!

discard the constant

Zeros: 23 2 2

So the zeros of f are the rational number +3/2 and the irrational numbers are and2 2

SOLUTION!!!

)2)(2)(23(2 xxx

Re-Cap of the Process

• Use Rational Zeros Theorem to locate possible zeros

• Use Calculator to narrow down possible zeros

• Use Synthetic Division to rewrite the function as (divisor)(quotient)

• Use the Zero Product Property to find all real zeros