polynomials

Polynomials

Objective: To review operations involving polynomials.

Definition

• Let be real numbers and let n be a nonnegative integer. A polynomial in x is an expression of the form

where . The polynomial is of degree n, is the leading coefficient, and is the constant term.

naaaa ,...,, 210

011

1 ,... axaxaxa nn

nn

0na na

0a

Polynomials

• A polynomial with one term is called a monomial.• A polynomial with two terms is called a binomial.• A polynomial with three terms is called a trinomial.

• In standard form, a polynomial is written with descending powers of x.

Example 1

Polynomial Standard form Degreea) 7

b) 2

c) 8 8 0

xxx 3254 72 2345 27 xxx

294 x 49 2 x

Operations with Polynomials

• You can add and subtract polynomials in much the same way you add and subtract real numbers. Simply add or subtract the like terms (terms having the same variables and the same powers) by adding or subtracting their coefficients.

Example 2

• Add or subtract the following polynomials.

• a) )82()375( 2323 xxxxx

Example 2

• Add or subtract the following polynomials.

• a)

• Put the like terms together and add their coefficients.

)82()375( 2323 xxxxx

)83()27()5( 2233 xxxxx

556 23 xxx

Example 2

• Add or subtract the following polynomials.• You Try

• b)

)343()247( 2424 xxxxxx

Example 2

• Add or subtract the following polynomials.• You Try

• b)

• Put the like terms together and add their coefficients.

)343()247( 2424 xxxxxx

2)34()4()37( 2244 xxxxxx

2734 24 xxx

Products

• To find the product of two polynomials, you can use the distributive method. If you are multiplying two binomials, you can also FOIL to find the answer.

Example 3

• Multiply the following binomials. )75)(23( xx

Example 3

• Multiply the following binomials.

• First, the distributive method.

)75)(23( xx

14102115)75(2)75(3 2 xxxxxx

141115 2 xx

Example 3

• Multiply the following binomials.

• First, the distributive method.

• Now, FOIL

)75)(23( xx

14102115)75(2)75(3 2 xxxxxx

141115 2 xx

14102115)75)(23( 2 xxxxx

141115 2 xx

Example 3

• Multiply the following binomials.• You Try.

)3)(22( xx

Example 3

• Multiply the following binomials.• You Try.

)3)(22( xx

6262 2 xxx

642 2 xx

Example 4

• When multiplying two trinomials, you must use the distributive method and collect like terms. It is easiest to do this using a vertical arrangement.

Example 4

• When multiplying two trinomials, you must use the distributive method and collect like terms. It is easiest to do this using a vertical arrangement.

• Multiply )13)(22( 22 xxxx

Example 4

• When multiplying two trinomials, you must use the distributive method and collect like terms. It is easiest to do this using a vertical arrangement.

• Multiply )13)(22( 22 xxxx

23422 3)13( xxxxxx xxxxxx 262..)13(2 232

262...............)13(2 22 xxxx

Example 4

• When multiplying two trinomials, you must use the distributive method and collect like terms. It is easiest to do this using a vertical arrangement.

• Multiply )13)(22( 22 xxxx

23422 3)13( xxxxxx xxxxxx 262..)13(2 232

262...............)13(2 22 xxxx

243 234 xxxx

Example 4

• Multiplying two trinomials.• You Try.

)22)(32( 22 xxxx

Example 4

• Multiplying two trinomials.• You Try.

)22)(32( 22 xxxx

23422 22)22( xxxxxx

xxxxxx 424........)22(2 232

636.............)22(3 22 xxxx

67632 234 xxxx

Example 5

• Multiplying the sum and difference of the same terms.

)4)(4( xx

Example 5

• Multiplying the sum and difference of the same terms.

)4)(4( xx

16442 xxx

162 x

Example 5

• Multiplying the sum and difference of the same terms.

)52)(52( xx

Example 5

• Multiplying the sum and difference of the same terms.

)52)(52( xx

2510104 2 xxx

254 2 x

Example 5

• Multiplying the sum and difference of the same terms.• You Try:

)63)(63( xx

Example 5

• Multiplying the sum and difference of the same terms.• You Try:

)63)(63( xx

3618189 2 xxx

369 2 x

Example 6

• Squaring a Binomial.

2)32( x

Example 6

• Squaring a Binomial.

2)32( x

9664)32)(32( 2 xxxxx

9124 2 xx

Example 6

• Squaring a Binomial.

2)43( x

Example 6

• Squaring a Binomial.

2)43( x

16)12(29 2 xx

16249 2 xx

Example 6

• Squaring a Binomial.• You Try:

2)24( x

Example 6

• Squaring a Binomial.• You Try:

2)24( x

4)8(216 2 xx

41616 2 xx

Example 7

• Cube of a Binomial.

32233 33)( yxyyxxyx

Example 7

• Cube of a Binomial.

32233 33)( yxyyxxyx

32233 33333)3( xxxx

27279)3( 233 xxxx

Example 7

• Cube of a Binomial.

32233 33)( yxyyxxyx

32233 44234)2(3)2()42( xxxx

6496488)42( 233 xxxx

Example 7

• Cube of a Binomial.• You Try:

32233 33)( yxyyxxyx

3)23( x

Example 7

• Cube of a Binomial.• You Try:

32233 33)( yxyyxxyx

32233 22332)3(3)3()23( xxxx

8365427)23( 233 xxxx

Homework

• Pages 29-30• 3-21 multiples of 3• 30-45 multiples of 3• 48-69 multiples of 3