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
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