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2.5 Zeros of Polynomial Functions

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2.5 Zeros of Polynomial Functions

Fundamental Theorem of Algebra

Rational Zero Test

Upper and Lower bound Rule

If f(x) is a polynomial of degree “n” > 0, then f(x) has at least one zero in the complex number system.

Complex zero’s (roots) come in pairs

If a + bi is a zero, then a – bi is a zero.

If f(x) is a polynomial of degree “n”>0, then there are as many zeros as degree.

If f(x) is a third degree function, then

f(x) = an(x – c1)(x – c2)(x – c3) where care complex numbers.

Complex zero’s (roots) come in pairs

If a + bi is a zero, then a – bi is a zero.

If f(x) has integer coefficients, then all possible zeros are

factors of the constant

factor of the lead coefficient

If f(x) has integer coefficients, then all possible zeros are

factors of the constant

factor of the lead coefficient

f(x) = x 3 – 7x 2 + 4x + 12

Possible zeros± 1, ± 2, ± 3, ± 4, ± 6, ± 12

± 1

- 1 |1-7412

-1 8 -12

1 - 812 0

So – 1 is a zero

How do you want to find the other zeros.

x 2 – 8x + 12

Let f(x) = anxn + an-1xn-1 + ….a1x + a0 ; with real coefficients and a0≠ 0.

Part 1

The number of positive real zeros equals (or a even number less), the number of variation in the sign of the coefficient (switching from positive to negative or negative to positive).

Let f(x) = anxn + an-1xn-1 + ….a1x + a0 ; with real coefficients and a0≠ 0.

Part 2

The number of negative real zeros equals (or a even number less), the number of variation in the sign of the coefficient

(switching from positive to negative or negative to positive) in f(- x).

f(x) = 4x3 - 3x2 +2x – 1

How many times does the sign change ?

f(x) = 4x3 - 3x2 +2x – 1

How many times does the sign change ?

3 times.

There are 3 or 1 positive zeros.

f(x) = 4x3 - 3x2 +2x – 1

What about f( -x) = -4x3 – 3x2 – 2x - 1

How many times does the sign change ?

f(x) = 4x3 - 3x2 +2x – 1

What about f( -x) = -4x3 – 3x2 – 2x - 1

How many times does the sign change ?

No change, no negative zeros.

If c > 0 ( “c” the number you divide by) and the last row of synthetic division is all positive or zero, the c| is the upper bound

So there is no zero larger then c, where c > 0.

If c < 0 and the last row alternate signs

( zero count either way), then c is the lower bound.

Check to see if 3 is the upper bound?

3|2- 512- 5All signs are 6 345positive.

2 11540

3 is an upper bound

Check to see if - 1 is the lower bound?

- 1|2- 512- 5All signs are -2 7-19switch.

2 - 719-24

-1 is an lower bound

Find the zeros

Page 160 – 164

# 5, 15, 23, 35,

42, 50, 57, 65,

73, 81, 85, 93,

103, 108, 111

Page 160 – 164

# 9, 19, 29, 41,

53, 61, 64, 77,

87, 97, 105,125

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