Let's solve each problem step by step. The task is to
multiply a monomial by a polynomial and simplify the expression.
We use the
distributive property:
$$
a(b + c) = ab + ac
$$
---
1) $ 9(6p + 4n) $
Distribute 9 to both terms:
$$
9 \cdot 6p = 54p, \quad 9 \cdot 4n = 36n
$$
Answer: $ \boxed{54p + 36n} $
---
2) $ 2s(4s - 9c) $
Distribute $2s$:
$$
2s \cdot 4s = 8s^2, \quad 2s \cdot (-9c) = -18sc
$$
Answer: $ \boxed{8s^2 - 18sc} $
---
3) $ 3c^2(7c^2 + 4c - 8) $
Distribute $3c^2$:
- $3c^2 \cdot 7c^2 = 21c^4$
- $3c^2 \cdot 4c = 12c^3$
- $3c^2 \cdot (-8) = -24c^2$
Answer: $ \boxed{21c^4 + 12c^3 - 24c^2} $
---
4) $ 6k^2(3k^2 - 7ks + 9s^2) $
Distribute $6k^2$:
- $6k^2 \cdot 3k^2 = 18k^4$
- $6k^2 \cdot (-7ks) = -42k^3s$
- $6k^2 \cdot 9s^2 = 54k^2s^2$
Answer: $ \boxed{18k^4 - 42k^3s + 54k^2s^2} $
---
5) $ 2d^2(6d^2 + 8dn + 9n^2) $
Distribute $2d^2$:
- $2d^2 \cdot 6d^2 = 12d^4$
- $2d^2 \cdot 8dn = 16d^3n$
- $2d^2 \cdot 9n^2 = 18d^2n^2$
Answer: $ \boxed{12d^4 + 16d^3n + 18d^2n^2} $
---
6) $ 4(7b^2 + 5b - 3) $
Distribute 4:
- $4 \cdot 7b^2 = 28b^2$
- $4 \cdot 5b = 20b$
- $4 \cdot (-3) = -12$
Answer: $ \boxed{28b^2 + 20b - 12} $
---
7) $ 4d(8d^2 - 2d + 9) $
Distribute $4d$:
- $4d \cdot 8d^2 = 32d^3$
- $4d \cdot (-2d) = -8d^2$
- $4d \cdot 9 = 36d$
Answer: $ \boxed{32d^3 - 8d^2 + 36d} $
---
8) $ 9c(4c + 3) $
Distribute $9c$:
- $9c \cdot 4c = 36c^2$
- $9c \cdot 3 = 27c$
Answer: $ \boxed{36c^2 + 27c} $
---
9) $ 8r^2(3r^2 - 5r + 6) $
Distribute $8r^2$:
- $8r^2 \cdot 3r^2 = 24r^4$
- $8r^2 \cdot (-5r) = -40r^3$
- $8r^2 \cdot 6 = 48r^2$
Answer: $ \boxed{24r^4 - 40r^3 + 48r^2} $
---
10) $ 2s(7s^2 - 3sh - 5h^2) $
Distribute $2s$:
- $2s \cdot 7s^2 = 14s^3$
- $2s \cdot (-3sh) = -6s^2h$
- $2s \cdot (-5h^2) = -10sh^2$
Answer: $ \boxed{14s^3 - 6s^2h - 10sh^2} $
---
✔ Final Answers:
1) $ 54p + 36n $
2) $ 8s^2 - 18sc $
3) $ 21c^4 + 12c^3 - 24c^2 $
4) $ 18k^4 - 42k^3s + 54k^2s^2 $
5) $ 12d^4 + 16d^3n + 18d^2n^2 $
6) $ 28b^2 + 20b - 12 $
7) $ 32d^3 - 8d^2 + 36d $
8) $ 36c^2 + 27c $
9) $ 24r^4 - 40r^3 + 48r^2 $
10) $ 14s^3 - 6s^2h - 10sh^2 $
Let me know if you'd like these written in a printable format or need help with any specific step!
Parent Tip: Review the logic above to help your child master the concept of multiplying polynomials by monomial worksheet.
