8-1 Adding and Subtracting Polynomials

Quick Review

A monomial is a number, a variable, or a product of a number and one or more variables. A polynomial is a monomial or the sum of two or more monomials. The degree of a polynomial in one variable is the same as the degree of the monomial with the greatest exponent. To add two polynomials, add the like terms of the polynomials. To subtract a polynomial, add the opposite of the polynomial.

Example

What is the difference of 3 x 3 − 7 x 2 + 5 3 x cubed , minus , 7 x squared , plus 5  and 2 x 2 − 9 x − 1 ? 2 x squared , minus 9 x minus 1 question mark

  ( 3 x 3 − 7 x 2 + 5 ) − ( 2 x 2 − 9 x − 1 )   = 3 x 3 − 7 x 2 + 5 − 2 x 2 + 9 x + 1   = 3 x 3 + ( − 7 x 2 − 2 x 2 ) + 9 x + ( 1 + 5 )   = 3 x 3 − 9 x 2 + 9 x + 6 table with 4 rows and 2 columns , row1 column 1 , , column 2 open , 3 x cubed , minus , 7 x squared , plus 5 close minus open , 2 x squared , minus 9 x minus 1 close , row2 column 1 , , column 2 equals , 3 x cubed , minus , 7 x squared , plus 5 minus , 2 x squared , plus 9 x plus 1 , row3 column 1 , , column 2 equals , 3 x cubed , plus open negative , 7 x squared , minus , 2 x squared , close plus 9 x plus open 1 plus 5 close , row4 column 1 , , column 2 equals , 3 x cubed , minus , 9 x squared , plus 9 x plus 6 , end table

Exercises

Write each polynomial in standard form. Then name each polynomial based on its degree and number of terms.

6. 4 r + 3 − 9 r 2 + 7 r 4 r plus 3 minus , 9 r squared , plus 7 r
7. 3 + b 3 + b 2 3 plus , b cubed , plus , b squared
8. 3 + 8 t 2 3 plus , 8 t squared
9. n 3 + 4 n 5 + n − n 3 n cubed , plus , 4 n to the fifth , plus n minus , n cubed
10. 7 x 2 + 8 + 6 x − 7 x 2 7 x squared , plus 8 plus 6 x minus , 7 x squared
11. p 3 q 3 p cubed , q cubed

Simplify. Write each answer in standard form.

12. ( 2 v 3 − v + 8 ) + ( − v 3 + v − 3 ) open , 2 v cubed , minus v plus 8 close plus open negative , v cubed , plus v minus 3 close
13. ( 6 s 4 + 7 s 2 + 7 ) + ( 8 s 4 − 11 s 2 + 9 s ) open , 6 s to the fourth , plus , 7 s squared , plus 7 close plus open , 8 s to the fourth , minus , 11 s squared , plus 9 s close
14. ( 4 h 3 + 3 h + 1 ) − ( − 5 h 3 + 6 h − 2 ) open , 4 h cubed , plus 3 h plus 1 close minus open negative , 5 h cubed , plus 6 h minus 2 close
15. ( 8 z 3 − 3 z 2 − 7 ) − ( z 3 − z 2 + 9 ) open , 8 z cubed , minus , 3 z squared , minus 7 close minus open , z cubed , minus , z squared , plus 9 close

8-2 Multiplying and Factoring

Quick Review

You can multiply a monomial and a polynomial using the Distributive Property. You can factor a polynomial by finding the greatest common factor (GCF) of the terms of the polynomial.

Example

What is the factored form of 10 y 4 − 12 y 3 + 4 y 2 ? 10 y to the fourth , minus , 12 y cubed , plus , 4 y squared , question mark

First find the GCF of the terms of the polynomial.

10 y 4 = 2 · 5 · y · y · y · y 12 y 3 = 2 · 2 · 3 · y · y · y 4 y 2 = 2 · 2 · y · y table with 3 rows and 2 columns , row1 column 1 , 10 y to the fourth , column 2 equals 2 middle dot 5 middle dot y middle dot y middle dot y middle dot y , row2 column 1 , 12 y cubed , column 2 equals 2 middle dot 2 middle dot 3 middle dot y middle dot y middle dot y , row3 column 1 , 4 y squared , column 2 equals 2 middle dot 2 middle dot y middle dot y , end table

The GCF is 2 · y · y  or  2 y 2 . 2 middle dot y middle dot y , or . 2 y squared , .

Then factor out the GCF.

10 y 4 − 12 y 3 + 4 y 2 = 2 y 2 ( 5 y 2 ) + 2 y 2 ( − 6 y ) + 2 y 2 ( 2 )   = 2 y 2 ( 5 y 2 − 6 y + 2 ) table with 2 rows and 2 columns , row1 column 1 , 10 y to the fourth , minus , 12 y cubed , plus , 4 y squared , column 2 equals , 2 y squared , open , 5 y squared , close plus , 2 y squared , open negative 6 y close plus , 2 y squared , open 2 close , row2 column 1 , , column 2 equals , 2 y squared , open , 5 y squared , minus 6 y plus 2 close , end table

Exercises

Simplify each product. Write in standard form.

16. 5 k ( 3 − 4 k ) 5 k open 3 minus 4 k close
17. 4 m ( 2 m + 9 m 2 − 6 ) 4 m open 2 m plus , 9 m squared , minus 6 close
18. 6 g 2 ( g − 8 ) 6 g squared , open g minus 8 close
19. 3 d ( 6 d + d 2 ) 3 d open 6 d plus , d squared , close
20. − 2 n 2 ( 5 n − 9 + 4 n 2 ) negative , 2 n squared , open 5 n minus 9 plus , 4 n squared , close
21. q ( 11 + 8 q − 2 q 2 ) q open 11 plus 8 q minus , 2 q squared , close

Find the GCF of the terms of each polynomial. Then factor the polynomial.

22. 12 p 4 + 16 p 3 + 8 p 12 p to the fourth , plus , 16 p cubed , plus 8 p
23. 3 b 4 − 9 b 2 + 6 b 3 b to the fourth , minus , 9 b squared , plus 6 b
24. 45 c 5 − 63 c 3 + 27 c 45 c to the fifth , minus , 63 c cubed , plus 27 c
25. 4 g 2 + 8 g 4 g squared , plus 8 g
26. 3 t 4 − 6 t 3 − 9 t + 12 3 t to the fourth , minus , 6 t cubed , minus 9 t plus 12
27. 30 h 5 − 6 h 4 − 15 h 3 30 h to the fifth , minus , 6 h to the fourth , minus , 15 h cubed
28. Reasoning The GCF of two numbers p and q is 5. Can you find the GCF of 6p and 6q? Explain your answer.

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