A polynomial function is a function of the form

f (x) = an x n + an – 1 x

n – 1 +· · ·+ a 1 x + a 0

Where an 0 and the exponents are all whole numbers.

A polynomial function is in standard form if its terms are written in descending order of exponents from left to right.

For this polynomial function, an is the leading coefficient,

a 0 is the constant term, and n is the degree.

an 0

an

an leading coefficient

a 0

a0 constant term n

n

degree

descending order of exponents from left to right.

n n – 1

OFFICIAL POLYNOMIAL JARGON

Let’s remember how to do long division.

7 8 4 03

2

6

1 8

6

-1 8

0 4

1

- 3

1 0

3

- 9

1

Remainder = 1

Solution is: 2613 31

So now, we’re going to use the same process to divide polynomials.

(x + 4) x3 + 6x2 + 2x + 10

x2

Use th

e 1s

t ter

m to

dete

rmin

e wha

t to

mul

tiply b

y.xx

tim

es w

hat eq

uals x

3 ?

x3

2x2

+ 4x2

x tim

es w

hat eq

uals 2

x2 ?

+ 2x

2x2

+ 2x+ 8x

- 6x + 10x

tim

es w

hat eq

uals –6x

?

- 6

- 6x - 24

34 This is the remainder!

Solution is: x2 + 2x – 6 +

434x

You try . . .

(x + 6) 2x3 + 15x2 + 20x + 10

2x2

x tim

es w

hat eq

uals 2

x3 ?

2x3

3x2

+ 12x2

x tim

es w

hat eq

uals 3

x2 ?

+ 3x

3x2+ 20x+ 18x

2x + 10x

tim

es w

hat eq

uals 2

x?

+ 2

2x + 12

-2 This is the remainder!

Solution is: 2x2 + 3x + 2 +

62

x

Another way to divide a polynomial is to use synthetic division.

Use synthetic division to divide

2 x 4 + 8 x

2 + 5 x 7 by (x – 3).

This is in the format (x – k).

So, k = 3

What is k if you’re dividing by (x + 7) ?-7

Using Synthetic Division

Polynomial in standard form

Using Synthetic Division

2 x 4 + 0 x

3 + (–8 x 2) + 5 x + (–7)

2 6

6

10

18

35

30 105

98

2 0 –8 5 –7 CoefficientsCoefficients

3

k-value

3 •

SOLUTION

Polynomial instandard form

3

98

x

Remainder Constant

+ 35

x-term

+ 10 x

X2-term

+ 6 x2

X3-term

+ 2 x3

Polynomial in standard form

3 x 4 + 4 x

3 + 0 x 2 + -8 x + -20

3 -2

-6

4

4

-16

-8 32

12

3 4 0 -8 -20 CoefficientsCoefficients

-2

k-value

-2 •

Divide 3x4 + 4x3 – 8x – 20 by (x + 2)

Polynomial instandard form

212x

Remainder Constant - 16

X-term

+ 4 x

X2-term

- 2 x2

X3-term

+ 3 x3

You Try . . .

A Few Notes

--The divisor must be of the form (x – k).

--If it is not, then you must use long division (i.e. 2x – 7).

--If the remainder is 0, then the answer is a factor of the original polynomial!

Divide x2 + 7x – 8 by (x – 1)

1 7 -81

1

1

8

8

0

Remainder

x-term constant

x + 8 (x – 1)(x + 8) = x2 + 7x – 8 !!

--Something really cool f(k) = the remainder !!

Try some more.

Divide. x2 - 2x - 8 (x - 4)

Divide. 3x3 - 2 x2 + 5x - 1 (x + 2)

Divide. 4x3 - 7x + 8 (2x - 1)