Section P.3

Polynomials and Special Products

Definition of a Polynomial in x1

1 1 0n n

n na x a x a x a

Examples: Find the leading coefficient and degree of each polynomial function.

Polynomial Function Leading Coefficient Degree5 3( ) 2 3 5 1f x x x x

3 2( ) 6 7f x x x x

( ) 14f x

-2 5

1 3

14 0

Definition of a Polynomial in xLet a0, a1, a2,…an be real numbers and let n be a

nonnegative integer.

A polynomial in x is an expression of the form

Where an ≠ 0.

• The polynomial is of degree n.

• The leading coefficient is an

• The constant term is a0.

11 1 0...n n

n na x a x a x a

Definition of a Polynomial in xLet a0, a1, a2,…an be real numbers and let n be a

nonnegative integer.

A polynomial in x is an expression of the form

Where an ≠ 0.

11 1 0...n n

n na x a x a x a

Is this a polynomial?

3 3x x1

3 2(3 )x x

Definition of a Polynomial in xLet a0, a1, a2,…an be real numbers and let n be a

nonnegative integer.

A polynomial in x is an expression of the form

Where an ≠ 0.

11 1 0...n n

n na x a x a x a

Is this a polynomial?

2 15x x 2 5x

x

Names of Polynomials

a) x2 – 16

b) x + 8

c) x4 + x3 – x – 15

d) 7

e) x2 – 6x + 5

f) x4y3 + 7

• 2nd degree, binomial

• 1st degree, binomial

• 4th degree, polynomial

• Constant, monomial

• 2nd degree, trinomial

• 7th degree, binomial

FOIL

)3( x )5( x

F O I Lx52x x3 15

Square a Binomial2)( ba ))(( baba

2a ab ab 2b22 2 baba

Now here is a shortcut worth knowing.

Square a Binomial2)( ba 22 2 baba

These are directions for a shortcut in binomial multiplication.

2)3( x a2 Square the first term

2x2ab Double the product

of the termsx)3(2 b2 Square the last term

23

962 xx

Square a Binomial Negative Version

2)( ba 22 2 baba These are directions for a shortcut in binomial multiplication.

2)4( x a2 Square the first term.

2x2ab Double the product

of the terms . Keep the signx)4(2 b2 Square the last term

Always positive.

24

1682 xx

Binomial Squared Examples

A) (t + u)2

B) (2m - p)2

C) (4p + 3q)2

D) (5r - 6s)2

E) (3k - 1/2)2

t2 + 2tu + u2

4m2 - 4mp + p2

16p2 + 24pq + 9q2

25r2- 60rs + 36s2

4

139 2 kk

The Product of the Sum and Difference of Two Terms

First start with two terms.

a bNow write two binomials.

)( ba )( ba One a sum The other a difference

2a ab ab 2b 22 ba

Binomial Sum and Difference Rule

• When multiplying two binomials that differ only in the sign between their terms, subtract the square of the last term from the square of the first term.

)( ba )( ba 22 ba Conjugate Factors = Difference of

Squares

Sum and Difference Examples

A) (6a + 3)(6a - 3)

B) (10m + 7)(10m - 7)

C) (7p + 2q)(7p - 2q)

D) (3r - 1/2)(3r + 1/2)

= 36a2 - 9

= 100m2 - 49

= 49p2 - 4q2

4

19 2 r

Cube a binomial3( )a b

3a

2( ) ( )a b a b 2 2( 2 )a ab b

22a b 2ab

( )a b

2a b 22ab 3b3 2 2 33 3a a b ab b

Apply the formula

3 3 2 2 3( ) 3 3a b a a b ab b

3( 2)x 3 2 2 33 (2) 3 (2) (2)x x x 3 26 12 8x x x

Homework

• Page 30• 1 – 6• 9 – 42 multiples of 3• 53, 59, 63, 71, 73, 86, 96, 97