Practice and Problem-Solving Exercises

A Practice

See Problems 1 and 2.

Expand each binomial.

8. ( x − y ) 3 open x minus y close cubed
9. ( a + 2 ) 4 open eh plus 2 close to the fourth
10. ( 6 + a ) 6 open 6 plus eh close to the sixth
11. ( x − 5 ) 3 open x minus 5 close cubed
12. ( y + 1 ) 8 open y plus 1 close to the eighth
13. ( x + 2 ) 10 open x plus 2 close to the tenth
14. ( b − 4 ) 7 open b minus 4 close to the seventh
15. ( b + 3 ) 9 open b plus 3 close to the ninth
16. ( 2 x − y ) 7 open 2 x minus y close to the seventh
17. ( a + 3 b ) 4 open eh plus 3 b close to the fourth
18. ( 4 x + 2 ) 6 open 4 x plus 2 close to the sixth
19. ( 4 − x ) 8 open 4 minus x close to the eighth
20. ( 4 x + 5 ) 2 open 4 x plus 5 , close squared
21. ( 3 a − 7 ) 3 open 3 eh minus 7 close cubed
22. ( 2 a + 16 ) 6 open 2 eh plus 16 close to the sixth
23. ( 3 y − 11 ) 4 open 3 y minus 11 close to the fourth

B Apply

24. Think About a Plan The side length of a cube is ( x 2 − 1 2 ) . open . x squared , minus , 1 half . close . .  Determine the volume of the cube.

    • Rewrite the binomial as a sum.
    • Consider ( a + b ) n . open eh plus b close to the n . .  Identify a and b in the given binomial.
    • Which row of Pascal's Triangle can be used to expand the binomial?
25. In the expansion of ( 2 m − 3 n ) 9 , open 2 m minus 3 n close to the ninth . comma  one of the terms contains m 3 . m cubed , .
    1. What is the exponent of n in this term?
    2. What is the coefficient of this term?

Find the specified term of each binomial expansion.

26. Fourth term of ( x + 2 ) 5 open x plus 2 close to the fifth
27. Third term of ( x − 3 ) 6 open x minus 3 close to the sixth
28. Third term of ( 3 x − 1 ) 5 open 3 x minus 1 close to the fifth
29. Fifth term of ( a + 5 b 2 ) 4 open eh plus 5 , b squared , close to the fourth
30. Reasoning Explain why the coefficients in the expansion of ( x + 2 y ) 3 open x plus 2 y close cubed  do not match the numbers in the 3rd row of Pascal's Triangle.
31. Compare and Contrast What are the benefits and challenges of using the Binomial Theorem when expanding ( 2 x + 3 ) 2 ? open 2 x plus 3 , close squared , question mark  Using FOIL? Which method would you choose when expanding ( 2 x + 3 ) 6 ? open 2 x plus 3 close to the sixth . question mark  Why?

Expand each binomial.

32. ( 2 x − 2 y ) 6 open 2 x minus 2 y close to the sixth
33. ( x 2 + 4 ) 10 open , x squared , plus 4 close to the tenth
34. ( x 2 − y 2 ) 3 open , x squared , minus , y squared , close cubed
35. ( a − b 2 ) 5 open eh minus , b squared , close to the fifth
36. ( 3 x + 8 y ) 3 open 3 x plus 8 y close cubed
37. ( 4 x − 7 y ) 4 open 4 x minus 7 y close to the fourth
38. ( 7 a + 2 y ) 10 open 7 eh plus 2 y close to the tenth
39. ( 4 x 3 + 2 y 2 ) 6 open 4 , x cubed , plus 2 , y squared , close to the sixth
40. ( 3 b − 36 ) 7 open 3 b minus 36 close to the seventh
41. ( 5 a + 2 b ) 3 open 5 eh plus 2 b close cubed
42. ( b 2 − 2 ) 8 open , b squared , minus 2 close to the eighth
43. ( − 2 y 2 + x ) 5 open negative 2 , y squared , plus x close to the fifth
44. Geometry The side length of a cube is given by the expression ( 2 x + 8 ) . open 2 x plus 8 close .  Write a binomial power for the area of a face of the cube and for the volume of the cube. Then use the Binomial Theorem to expand and rewrite the powers in standard form.
45. Writing Explain why the terms of ( x − y ) n open x minus y close to the n  have alternating positive and negative signs.

46. Error Analysis A student expands ( 3 x − 8 ) 4 open 3 x minus 8 close to the fourth  as shown below. Describe and correct the student's error.

    
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