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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.
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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
