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Notes Day 6.2. Irrational Root Theorem Imaginary Root Theorem Factors and their multiplicity Find relative max/min on the graphing calculator. Notes 6.2 Irrational Root Theorem. If is an S / R / Z / x-intercept, then the conjugate

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Notes day 6 2

Notes Day 6.2

Irrational Root Theorem

Imaginary Root Theorem

Factors and their multiplicity

Find relative max/min on the

graphing calculator


Notes 6 2 irrational root theorem
Notes 6.2 Irrational Root Theorem

If is an S / R / Z / x-intercept, then the conjugate

__________ must ALSO be an S / R / Z / x-intercept

EXAMPLE

A polynomial equation with integer coefficients has 4 roots.

If two of them are and , what are the other two?

__________________ and _________________



Imaginary root theorem
Imaginary Root Theorem and

If a + bi is an S / R / Z / x-intercept, then the conjugate

__________ must ALSO be an S / R / Z / x-intercept

EXAMPLE

A polynomial equation with integer coefficients has 4 roots.

If two of them are 3 – i and 2i , what are the other two?

______________________ and _____________________


Write a cubic polynomial equation that has zeros at 2 and 4 i
Write a cubic Polynomial equation that has zeros at 2 and 4+i

ix

-4x

x2

-8x2

17x

x3

-4x

16

-4i

-ix

4i

-2x2

16x

-i2

-34


Multiplicity
Multiplicity and 4+i

The number of times an S / R / Z / x-intercept is

___________________ in a given polynomial equation.

REPEATED

EXAMPLES: name the root(s) and multiplicity

y = x(x2 – 4x + 4)

y = x(x-2)(x-2)

Once

2

Once

-1

0

Once

2

Twice

Twice

-4


Solve the equation
Solve the equation! and 4+i

The square root method works since no linear term

x2+ 4 = 0

x2 = – 4




Factor to find roots but use calculator to find relative min/max.

y = 2x3 + 14x2 – 3x – 21

y-int =

-21


Factor to find roots but use calculator to find relative min/max.

y = 3x5 + 18x4 + 27x3

Work this on back of your notes!

Local minimum is:

-25


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