finding all real zeros of a polynomial with examples
TRANSCRIPT
Finding all real zeros
of a Polynomial
Find all the real zeros of
2,1
842,1
814872)( 234 xxxxxf
Use the Rational Zeros Theorem to make a list of possible rational zeros
2
1,8,4,2,1
Find all the real zeros of 814872)( 234 xxxxxf
?2
1x
?4x
Use your graphing calculator to narrow down the possible rational zeros
the function seems to cross the x axis at these points…..
Find all the real zeros of
82
114
2
18
2
17
2
12)
2
1(
234
f
814872)( 234 xxxxxf
Use the remainder and factor theorems to test the possible zeros
0
8414484742)4( 234 f
0
since the remainder is zero, x + ½ is a factor!
since the remainder is zero, x - 4 is a factor!
Find all the real zeros of 814872)( 234 xxxxxf
Use one of the divisors to divide the dividend
4 2 -7 -8 14 8
2 1 -4 -2 0
8 4 -16 -8
Let’s start with (x - 4)
So the dividend is equal to:
)2412)(4( 23 xxxx
Find all the real zeros of
Now, let’s use the other factor of (x + ½)
to divide the second factor:
)2412)(4( 23 xxxx
-1 0 2
2 1 -4 -2
2 0 -4 0
2
1
So the dividend is equal to: )42)(21( 2 xx
Which means our original function is equal to: )42)(2
1)(4( 2 xxx
Find all the real zeros of
Synthetic division has allowed us to factor most of this polynomial, but now we can use other factor techniques to take care of the rest!
)42)(21)(4( 2 xxx
814872)( 234 xxxxxf
Factor out the GCF
And then use difference of two squares method to factor one last time
)2)(21)(4(2 2 xxx
)2)(2)(21)(4(2 xxxx
Find all the real zeros of 814872)( 234 xxxxxf
Now that you have the polynomial in factored form, find those zeros!!!
)2)(2)(21)(4(2 xxxx
discard the constant
Zeros: 4 21 2 2
So the zeros of f are the rational numbers 4 and -1/2 and the irrational numbers are and2 2
SOLUTION!!!
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)
• Repeat Synthetic Division of quotient until you can factor the remaining quotient
• Use the Zero Product Property to find all real zeros