1 Equivalence

You can represent algebraic expressions in many ways. When you add, subtract, multiply, divide, and factor polynomials, you replace one expression with an equivalent expression.

Adding and Subtracting Polynomials (Lesson 8-1)

( 3 x 2 + 4 x + 1 ) + ( 2 x 2 + 5 x + 8 ) = ( 3 x 2 + 2 x 2 ) + ( 4 x + 5 x ) + ( 1 + 8 ) = 5 x 2 + 9 x + 9 table with 3 rows and 1 column , row1 column 1 , open . 3 , x squared , plus 4 x plus 1 . close . plus . open . 2 , x squared , plus 5 x plus 8 . close , row2 column 1 , equals . open . 3 , x squared , plus 2 , x squared . close . plus . open . 4 x plus 5 x . close . plus . open , 1 plus 8 , close , row3 column 1 , equals 5 , x squared , plus 9 x plus 9 , end table

Multiplying Special Cases Lesson 8-4)

( ( 2 x + 3 ) ( 2 x − 3 ) = 4 x 2 − 9 open open 2 x plus 3 close open 2 x minus 3 close equals , 4 x squared , minus 9

Multiplying Binomials (Lesson 8-3)

( m + 4 ) ( 2 m − 5 ) = 2 m 2 − 5 m + 8 m − 20   = 2 m 2 + 3 m − 20 table with 2 rows and 2 columns , row1 column 1 , open , m plus 4 , close . open , 2 m minus 5 , close , column 2 equals 2 , m squared , minus 5 m plus 8 m minus 20 , row2 column 1 , , column 2 equals 2 , m squared , plus 3 m minus 20 , end table

2 Properties

The properties of real numbers are the basis of the laws of algebra. You can apply properties of real numbers, such as the Distributive Property, to polynomials.

Factoring Trinomials (Lessons 8-5 and 8-6) x 2 − 6 x + 8 = ( x − 2 ) ( x − 4 ) x squared , minus 6 x plus 8 equals open x minus 2 close open x minus 4 close

Factoring Special Cases (Lesson 8-7) 49 p 2 − 16 = ( 7 p + 4 ) ( 7 p − 4 ) 49 p squared , minus 16 equals open 7 p plus 4 close open 7 p minus 4 close

Factoring by Grouping (Lesson 8-8)

3 x 2 − 10 x − 8 = 3 x 2 − 12 x + 2 x − 8   = ( 3 x 2 − 12 x ) + ( 2 x − 8 )   = 3 x ( x − 4 ) + 2 ( x − 4 )   = ( 3 x + 2 ) ( x − 4 ) table with 4 rows and 2 columns , row1 column 1 , 3 , x squared , minus 10 x minus 8 , column 2 equals 3 , x squared , minus 12 x plus 2 x minus 8 , row2 column 1 , , column 2 equals . open . 3 , x squared , minus 12 x . close . plus . open , 2 x minus 8 , close , row3 column 1 , , column 2 equals 3 x . open , x minus 4 , close . plus 2 . open , x minus 4 , close , row4 column 1 , , column 2 equals . open , 3 x plus 2 , close . open , x minus 4 , close , end table