Problem 5 Subtracting Polynomials

What is a simpler form of ( x 3 − 3 x 2 + 5 x ) − ( 7 x 3 + 5 x 2 − 12 ) ? x cubed , minus , 3 x squared , plus 5 x close minus open , 7 x cubed , plus , 5 x squared , minus 12 close question mark

Method 1 Subtract vertically.

x 3 − 3 x 2 + 5 x     − ( 7 x 3 + 5 x 2     − 12 ) x 3 − 3 x 2 + 5 x     − 7 x 3 − 5 x 2     + 12 − 6 x 3 − 8 x 2 + 5 x + 12 Line up like terms .   Then add the opposite of each term in the polynomial being subtracted .     table with 1 row and 2 columns , row1 column 1 , table with 5 rows and 7 columns , row1 column 1 , x cubed , column 2 minus , column 3 3 x squared , column 4 plus , column 5 5 x , column 6 , column 7 , row2 column 1 , negative open , 7 x cubed , column 2 plus , column 3 5 x squared , column 4 , column 5 , column 6 minus , column 7 12 close , row3 column 1 , x cubed , column 2 minus , column 3 3 x squared , column 4 plus , column 5 5 x , column 6 , column 7 , row4 column 1 , negative , 7 x cubed , column 2 minus , column 3 5 x squared , column 4 , column 5 , column 6 plus , column 7 12 , row5 column 1 , negative , 6 x cubed , column 2 minus , column 3 8 x squared , column 4 plus , column 5 5 x , column 6 plus , column 7 12 , end table , column 2 table with 5 rows and 1 column , row1 column 1 , cap lineupliketerms . . , row2 column 1 , , row3 column 1 , table with 2 rows and 1 column , row1 column 1 , cap thenaddtheoppositeofeachtermin , row2 column 1 , thepolynomialbeingsubtracted . . , end table , row4 column 1 , , row5 column 1 , , end table , end table

Method 2 Subtract horizontally.

( x 3 − 3 x 2 + 5 x ) − ( 7 x 3 + 5 x 2 − 12 )   = x 3 − 3 x 2 + 5 x − 7 x 3 − 5 x 2 + 12 Write the opposite of each term in the polynomial being subtracted. = ( x 3 − 7 x 3 ) + ( − 3 x 2 − 5 x 2 ) + 5 x + 12 Group like terms. = − 6 x 3 − 8 x 2 + 5 x + 12 Simplify. table with 4 rows and 2 columns , row1 column 1 , open , x cubed , minus , 3 x squared , plus 5 x close minus open , 7 x cubed , plus , 5 x squared , minus 12 close , column 2 , row2 column 1 , equals , x cubed , minus , 3 x squared , plus 5 x minus , 7 x cubed , minus , 5 x squared , plus 12 , column 2 cap writetheoppositeofeachterminthepolynomialbeingsubtracted. , row3 column 1 , equals open , x cubed , minus , 7 x cubed , close plus open negative , 3 x squared , minus , 5 x squared , close plus 5 x plus 12 , column 2 cap groupliketerms. , row4 column 1 , equals negative , 6 x cubed , minus , 8 x squared , plus 5 x plus 12 , column 2 cap simplify. , end table

✓ Got It?

5. What is a simpler form of ( − 4 m 3 − m + 9 ) − ( 4 m 2 + m − 12 ) ? open negative , 4 m cubed , minus m plus 9 close minus open , 4 m squared , plus m minus 12 close question mark

✓ Lesson Check

Do you know HOW?

Find the degree of each monomial.

1. − 7 x 4 negative , 7 x to the fourth
2. 8 y 2 z 3 8 y squared . z cubed

Simplify each sum or difference.

3. ( 5 r 3 + 8 ) + ( 6 r 3 + 3 ) open , 5 r cubed , plus 8 close plus open , 6 r cubed , plus 3 close
4. ( x 2 − 2 ) − ( 3 x + 5 ) open , x squared , minus 2 close minus open 3 x plus 5 close

Do you UNDERSTAND?

Vocabulary Name each polynomial based on its degree and number of terms.

5. 5 x 2 + 2 x + 1 5 x squared , plus 2 x plus 1
6. 3 z − 2 3 z minus 2
7. Compare and Contrast How are the processes of adding monomials and adding polynomials alike? How are the processes different?