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100 100 100 100 100 100 300 300 300 300 200 200 200 200 200 400 300 500 500 400 400 400 400 500 500 500 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds
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100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

Dec 22, 2015

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Page 1: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

100 100 100100 100 100

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400 400400 400

500 500500 500

400

300

200

Long Division

RemainderTheorem

Factor Theorem

SyntheticDivision

Rational Zeros

Theorem

Upper &Lower

Bounds

Page 2: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

Division involves a dividend divided by a divisor to obtain a

quotient and a _________.

remainder

Page 3: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

What is the answer of 2x4-x3-2 divided by 2x2+x+1? Write in

polynomial form.

2x4-x3-x=(2x2+x+1)(x2-2)+(x-2)

Page 4: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

Divide f(x)=x2-2x+3 b x-1. Write in polynomial form.

f(x)=(x=1)2+2

Page 5: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

f(x)=(x2+x+4)(x+3)-21

Divide f(x)=x3+4x2+7x-9 by x+3 and write a summary statement

in polynomial form.

Page 6: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

Divide f(x)=x4-2x3+3x2-4x+6 by x2+2x-, write a summary

statement in fraction form.

f(x)=(x2-4x+12)(x2+2x-1)-32x+18

Page 7: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

If a polynomial is divided by ___, then the remainder is

r=f(k).

x-k

Page 8: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

Using the remainder theorem, find the remainder when

f(x)=3x2+7x-20 is divided by x+1.

r=f(-1)=3(-1)2+7(-1)-20=3-7-20=-24

Page 9: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

What is the remainder when f(x)=2x2-3x+1 is divided by x-2?

3

Page 10: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

Use the remainder theorem to find the remainder when you divide f(x)=x3-x2+2x-1 by x+3

-43

Page 11: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

5

Use the remainder theorem to find the remainder when f(x)=2x3-3x2+4x-7 by x-2

Page 12: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

0

A polynomial function f(x) has a factor x-k if and only if f(k)=_.

Page 13: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

(3x-5)

Using the factor theorem, factor f(x)=3x2+7x-20 by dividing it by

the known factor x+4.

Page 14: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

Yes

Is x-1 a factor of x3-x2+x-1?

Page 15: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

No

Use the factor theorem to determine whether x-2 is a

factor of x3+3x-4

Page 16: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

Yes

Use the factor theorem to determine where x+2 is a factor

of 4x3+9x2-3x-10

Page 17: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

linear divisor

This shortcut method for the division of a polynomial by a

______ _______ x-k is synthetic division.

Page 18: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

2x3-3x2-5x-12 = 2x2+3x+4

x-3

Divide 2x3-3x2-5x-12 by x-3 using synthetic division and

write a summary statement in fraction form.

Page 19: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

X2-6x+9+ -11

x+1

Divide using synthetic division and write a summary statement

in fraction form: x3-5x2+3x-2

x+1

Page 20: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

9x2+97x+967+ 9670

x-10

Divide using synthetic division and write a summary statement

in fraction form: 9x3+7x2-3x

x-10

Page 21: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

-5x3-20x2-80x-317+ -1269

4-x

Divide using synthetic division and write a summary statement

in fraction form: 5x4-3x+1

4-x

Page 22: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

rational, irrational

Real zeros of a polynomial function are either ________

zeros – zeroes that are rational numbers – or __________

zeros – zeros that are irrational numbers.

Page 23: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

f(1)=(1)3-3(1)2+1=-1

f(-1)=(-1)3-3(-1)2+1=-3

So f has no rational zeros.

Find the rational zeros of f(x)=x3-3x2+1

Page 24: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

1

1, 2, 3, 6

(all positive & negative)

Use the rational zeroes theorem to write a list of all possible

rational zeroes of f(x)=6x3-5x-1

Page 25: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

1, 3, 9

1, 2

(all positive & negative)

Use the rational zeroes theorem to write a list of possible rational

zeroes:f(x)=2x3-x2-9x+9

Page 26: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

-1/2 and 4; rational

No irrational

Find all the zeroes of the function and identify each zero

as rational or irrational:f(x)=2x4-7x3-2x2-7x-4

Page 27: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

bounds

We narrow our search for real zeros by using a test that identifies upper and lower

______ for real zeros.

Page 28: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

-2

Find a lower bound of f(x)=2x4-7x3-8x2+14x+8

Page 29: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

Yes

Use synthetic division to prove that to number k is an upper

bound for the real zeroes of the function:

k=3; f(x)=2x3-4x2+x+3

Page 30: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

Yes

Use synthetic division to prove that the number k is an upper

bound for the real zeroes of the function f:

k=2; f(x)=x4-x3+x2+x-12

Page 31: 100 300 200 400 300 500 400 500 400 300 200 Long Division Remainder Theorem Factor Theorem Synthetic Division Rational Zeros Theorem Upper & Lower Bounds.

Yes

Use synthetic division to prove that the number k is a lower

bound for the real zeroes of the function f:

k=-1; f(x)=3x2-4x2+x-2


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