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# 2.5 Zeros of Polynomial Functions - PowerPoint PPT Presentation

2.5 Zeros of Polynomial Functions. Fundamental Theorem of Algebra Rational Zero Test Upper and Lower bound Rule. Fundamental Theorem of Algebra. 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

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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

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

factors of the constant

f(x) = x 3 – 7x 2 + 4x + 12

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

± 1

f(x) = x 3 – 7x 2 + 4x + 12Possible zeros ± 1, ± 2, ± 3, ± 4, ± 6, ± 12 ± 1

- 1 | 1 -7 4 12

-1 8 -12

1 - 8 12 0

So – 1 is a zero

How do you want to find the other zeros.

x 2 – 8x + 12

Find the zeros f(x) = 3x3 – x2 + 6x - 2

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.

f(x) = 2x3 – 5x2 + 12x - 5

Check to see if 3 is the upper bound?

3| 2 - 5 12 - 5 All signs are 6 3 45 positive.

2 1 15 40

3 is an upper bound

f(x) = 2x3 – 5x2 + 12x - 5

Check to see if - 1 is the lower bound?

- 1| 2 - 5 12 - 5 All signs are -2 7 -19 switch.

2 - 7 19 -24

-1 is an lower bound

f(x) = 2x3 – 5x2 + 12x - 5

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