This document discusses multiplying polynomials by monomials. It begins with an essential question about how to multiply polynomials by monomials and where this concept is applied. It then provides examples of multiplying polynomials by monomials step-by-step. The examples are worked out in detail, showing the multiplication at each step. It concludes with an example of writing and simplifying expressions for the area of different geometric figures.

1. SECTION 9-4
Multiply a Polynomial by a Monomial

2. ESSENTIAL QUESTION
• How do you multiply polynomials by monomials?
• Where you’ll see this:
• Travel, part-time job, sports, finance, geography

3. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )

4. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )

5. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2
4x
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )

6. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2
4x
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )

7. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2
4x +2xy
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )

8. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2
4x +2xy
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )

9. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2 3
4x +2xy 10w
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )

10. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2 3
4x +2xy 10w
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )

11. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2 3 2
4x +2xy 10w −20w
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )

12. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2 3 2
4x +2xy 10w −20w
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )

13. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2 3 2
4x +2xy 10w −20w
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )
3
−6n

14. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2 3 2
4x +2xy 10w −20w
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )
3
−6n

15. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2 3 2
4x +2xy 10w −20w
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )
3 2
−6n −8n

16. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2 3 2
4x +2xy 10w −20w
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )
3 2
−6n −8n

17. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2 3 2
4x +2xy 10w −20w
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )
3 2
−6n −8n +10n

18. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2 3 2
4x +2xy 10w −20w
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )
3 2
−6n −8n +10n

19. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2 3 2
4x +2xy 10w −20w
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )
3 3
3
−6n −8n +10n2
10x y

20. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2 3 2
4x +2xy 10w −20w
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )
3 3
3
−6n −8n +10n2
10x y

21. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2 3 2
4x +2xy 10w −20w
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )
3 3 2 5
3
−6n −8n +10n2
10x y +30x y

22. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2 3 2
4x +2xy 10w −20w
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )
3 3 2 5
3
−6n −8n +10n2
10x y +30x y

23. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2 3 2
4x +2xy 10w −20w
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )
3 3 2 5 4 4
3
−6n −8n +10n2
10x y +30x y +15x y

24. EXAMPLE 1
Simplify.
a. 2x(2x + y) 2
b. − 5w(−2w + 4w )
2 3 2
4x +2xy 10w −20w
2 2 2 3 2 2
c. − 2n(3n + 4n − 5) d. 5x y (2xy + 6y + 3x y )
3 3 2 5 4 4
3
−6n −8n +10n2
10x y +30x y +15x y
4 4 3 3 2 5
15x y +10x y + 30x y

25. EXAMPLE 2
Write and simplify an expression for the area of each figure.
a. 4h b.
3x2 + 2
6h + 1 2x2

26. EXAMPLE 2
Write and simplify an expression for the area of each figure.
a. 4h b.
3x2 + 2
6h + 1 2x2
4h(6h +1)

27. EXAMPLE 2
Write and simplify an expression for the area of each figure.
a. 4h b.
3x2 + 2
6h + 1 2x2
4h(6h +1)

28. EXAMPLE 2
Write and simplify an expression for the area of each figure.
a. 4h b.
3x2 + 2
6h + 1 2x2
4h(6h +1)
2
24h

29. EXAMPLE 2
Write and simplify an expression for the area of each figure.
a. 4h b.
3x2 + 2
6h + 1 2x2
4h(6h +1)
2
24h

30. EXAMPLE 2
Write and simplify an expression for the area of each figure.
a. 4h b.
3x2 + 2
6h + 1 2x2
4h(6h +1)
2
24h +4h

31. EXAMPLE 2
Write and simplify an expression for the area of each figure.
a. 4h b.
3x2 + 2
6h + 1 2x2
4h(6h +1)
24h 2
+4h units2

32. EXAMPLE 2
Write and simplify an expression for the area of each figure.
a. 4h b.
3x2 + 2
6h + 1 2x2
2 2
4h(6h +1) 2x (3x + 2)
24h 2
+4h units2

33. EXAMPLE 2
Write and simplify an expression for the area of each figure.
a. 4h b.
3x2 + 2
6h + 1 2x2
2 2
4h(6h +1) 2x (3x + 2)
24h 2
+4h units2

34. EXAMPLE 2
Write and simplify an expression for the area of each figure.
a. 4h b.
3x2 + 2
6h + 1 2x2
2 2
4h(6h +1) 2x (3x + 2)
4
24h 2
+4h units2 6x

35. EXAMPLE 2
Write and simplify an expression for the area of each figure.
a. 4h b.
3x2 + 2
6h + 1 2x2
2 2
4h(6h +1) 2x (3x + 2)
4
24h 2
+4h units2 6x

36. EXAMPLE 2
Write and simplify an expression for the area of each figure.
a. 4h b.
3x2 + 2
6h + 1 2x2
2 2
4h(6h +1) 2x (3x + 2)
4 2
24h 2
+4h units2 6x +4x

37. EXAMPLE 2
Write and simplify an expression for the area of each figure.
a. 4h b.
3x2 + 2
6h + 1 2x2
2 2
4h(6h +1) 2x (3x + 2)
4 2
24h 2
+4h units2 6x +4x units2

38. HOMEWORK

39. HOMEWORK
p. 392 #1-54 multiples of 3
“A candle loses none of its light by lighting another candle.”
- Unknown