Math-2

Section 1-6Add, subtract, multiply

Polynomials

VocabularyPolynomial: an expression or an equation formed by the

sum of “same-base” powers.

Terms of a polynomial: the individual powers that are

being added.

Zero function: horizontal line at y=0

constant function: horizontal line at y = ‘a’

linear function: slope = ‘a’, y-intercept = ‘b’

VocabularyDegree of a Polynomial: the largest exponent of the

polynomial.

Vocabulary

Standard form Polynomial: is written so the powers

have smaller exponents from left to right.

4444 35 xxxSimplify: means

add and subtract (if there are “like terms”),

multiply (parentheses implies multiplication), and

divide (for polynomials this will be covered in Math-3)

Degree of a polynomial: the biggest exponent (largest

power) of the polynomial

Simplifying Polynomials

1523 4545 xxxx

Combine “like” terms.

1523 4455 xxxxWhat property allowed us to “rearrange the

order” of the terms?

Commutative Property (of addition)

(combine “like” terms is English for addition).

174 45 xx

Your Turn: simplify

)52()442( 535 xxxxx

1.

2.

xxxxx 64423 535

If you add/subtract polynomials you get a sum

the is also a polynomial.

Polynomials are “closed” for

addition/subtraction!!!

xxx 102 35

4444 35 xxx

Multiplying Polynomials

2(3x – 1) Distributive property

6x - 2

Your Turn: simplify

)75(2 23 xxx

3.

4.

)2(3 2 xx xx 63 2

34 1410 xx

If you multiply polynomials you get a

polynomial as the product.

Polynomials are “closed” for multiplication!!!

Multiplying Polynomials

(x + 3)(2x – 1) Distributive property (twice)

)12( xx )12(3 x

22x x x6 3 Combine “like” terms

352 2 xx

Your Turn: simplify

)4)(52( xx

5.

6.

)3)(4( xx 122 xx

2032 2 xx

Multiplying PolynomialsHow do you multiply 2 * 3 * 4

three numbers?

6 * 4 = 24OR: 2 * 3 * 4

122 * = 24 OR: 2 * 3 * 4

8 * 3 = 24

Pick 2 factors and multiply them to get a product,

then multiply the product by the last factor

New Property

Associative Property (of addition): if you have 3

or more addends, pick two, add them 1st

2+3+4

(2+3)+4

(to visually show that we are picking

2, we group, or associate them

together with parentheses.

New Property

Associative Property (of multiplication): if you

have 3 or more factors, pick two, multiply them 1st

2*3*4

(2*3)*4

(to visually show that we are picking 2, we group,

or associate them together with parentheses).

)42( 2 xxx

Distributive property

(twice)

)42(3 2 xx

3x 22x x423x

simplify3x

)42)(3( 2 xxx

x6 12

2x x10

(x – 1)(2x + 3)(3x – 2) = ?

= [ (x – 1)(2x + 3) ] (3x – 2)

)23)(3322( 2 xxxx

)23)(32( 2 xxx

How do you multiply three binomials?

Pick 2 factors, multiply them to get a product, then

multiply the product by the last factor associative property.

)23()32(1)32( xxxx

“Box Method”

?)32)(23( 2 xxx

x3

)3(

)2(

“break apart” into individual terms

(row and column “headers”)

22x x

“Box Method”

?)32)(23( 2 xxx

x3

)3(

)2(

“multiply rows and colums”

22x x

36x 23x x9

“Box Method”

?)32)(23( 2 xxx

x3

)3(

)2(

“multiply rows and colums”

22x x

36x 23x x9

24x x2 6

“Box Method”

?)32)(23( 2 xxx

add (combine “like terms”)

36x 23x x9

24x x2 6

Diagonals have “like terms”

36x 2x x11 6

8. (Hint: use the box method)

)123)(123( 22 xxxx

Simplify

23x

x2

)1(

)1(x223x49x 36x

23x36x 24x x2

23x x2 1

22 )123( xx

)123)(123( 22 xxxx

49x 36x 23x36x 24x x2

23x x2 1

49x312x 22x x4 1

add (combine “like terms”)

Your Turn: Multiply

)42)(42( xx

9.

10.

)2)(2( xx

11. ))(( yxyx

42 x

164 2 x

22 yx

Your turn:

12.

))(( yxyx

)3)(3( xx

13.

14. 2)5( x

962 xx

25102 xx

22 2 yxyx

Your Turn: Multiply

)1)(1( 22 xx15.

16. )24)(24( 22 xx

3)1( x17.

12 24 xx

416 4 x

133 23 xxx