pre - algebra : multiplying polynomials
name
date
period
1. $3n(6n - 1)$
2. $3m(8m - 1)$
3. $5k(4k - 7)$
4) $2(7b - 8)$
5. $5n(4n + 6)$
6. $3(6n + 5)$
7. $2b(3b - 3)$
8. $8(8n - 2)$
9. $3(8p + 5)$
10. $4b(4b - 6)$
11. $3(3n - 4)$
12. $2(6x - 7)$
13. $3(5n - 2)$
14. $5(3x - 1)$
15. $6x^2(3x - 6)$
16. $7(8x + 2)$
17. $4r(r + 1)$
18. $2x^2(5x + 3)$
19. $5p(6p - 6)$
20. $3v(3v + 4)$

Question

Accepted Answer

Let's solve problem 1: $ 3n(6n - 1) $

# Explanation:
## Step 1: Apply the distributive property (multiplication over subtraction)
The distributive property states that $ a(b - c) = ab - ac $. Here, $ a = 3n $, $ b = 6n $, and $ c = 1 $. So we multiply $ 3n $ by each term inside the parentheses.
$ 3n \times 6n - 3n \times 1 $

## Step 2: Simplify each product
For the first product: $ 3n \times 6n = 18n^2 $ (using the rule $ a^m \times a^n = a^{m + n} $, here $ n \times n = n^{1 + 1}=n^2 $ and $ 3\times6 = 18 $)
For the second product: $ 3n \times 1 = 3n $

So putting it together, we get $ 18n^2 - 3n $

# Answer:
$ 18n^2 - 3n $

Let's solve problem 2: $ 3m(8m - 1) $

# Explanation:
## Step 1: Apply the distributive property
Using $ a(b - c)=ab - ac $ with $ a = 3m $, $ b = 8m $, $ c = 1 $
$ 3m\times8m - 3m\times1 $

## Step 2: Simplify each product
First product: $ 3m\times8m = 24m^2 $ ( $ 3\times8 = 24 $ and $ m\times m=m^2 $)
Second product: $ 3m\times1 = 3m $

So the result is $ 24m^2 - 3m $

# Answer:
$ 24m^2 - 3m $

Let's solve problem 3: $ 5k(4k - 7) $

# Explanation:
## Step 1: Apply the distributive property
$ 5k\times4k - 5k\times7 $ (using $ a(b - c)=ab - ac $ with $ a = 5k $, $ b = 4k $, $ c = 7 $)

## Step 2: Simplify each product
First product: $ 5k\times4k = 20k^2 $ ( $ 5\times4 = 20 $ and $ k\times k = k^2 $)
Second product: $ 5k\times7 = 35k $

So we have $ 20k^2 - 35k $

# Answer:
$ 20k^2 - 35k $

Let's solve problem 4: $ 2(7b - 8) $

# Explanation:
## Step 1: Apply the distributive property
$ 2\times7b - 2\times8 $ (using $ a(b - c)=ab - ac $ with $ a = 2 $, $ b = 7b $, $ c = 8 $)

## Step 2: Simplify each product
First product: $ 2\times7b = 14b $
Second product: $ 2\times8 = 16 $

So the result is $ 14b - 16 $

# Answer:
$ 14b - 16 $

Let's solve problem 5: $ 5n(4n + 6) $

# Explanation:
## Step 1: Apply the distributive property (multiplication over addition)
The distributive property for addition is $ a(b + c)=ab + ac $. Here, $ a = 5n $, $ b = 4n $, $ c = 6 $
$ 5n\times4n + 5n\times6 $

## Step 2: Simplify each product
First product: $ 5n\times4n = 20n^2 $ ( $ 5\times4 = 20 $ and $ n\times n = n^2 $)
Second product: $ 5n\times6 = 30n $

So we get $ 20n^2 + 30n $

# Answer:
$ 20n^2 + 30n $

Let's solve problem 6: $ 3(6n + 5) $

# Explanation:
## Step 1: Apply the distributive property
Using $ a(b + c)=ab + ac $ with $ a = 3 $, $ b = 6n $, $ c = 5 $
$ 3\times6n + 3\times5 $

## Step 2: Simplify each product
First product: $ 3\times6n = 18n $
Second product: $ 3\times5 = 15 $

So the result is $ 18n + 15 $

# Answer:
$ 18n + 15 $

Let's solve problem 7: $ 2b(3b - 3) $

# Explanation:
## Step 1: Apply the distributive property
$ 2b\times3b - 2b\times3 $ (using $ a(b - c)=ab - ac $ with $ a = 2b $, $ b = 3b $, $ c = 3 $)

## Step 2: Simplify each product
First product: $ 2b\times3b = 6b^2 $ ( $ 2\times3 = 6 $ and $ b\times b = b^2 $)
Second product: $ 2b\times3 = 6b $

So we have $ 6b^2 - 6b $

# Answer:
$ 6b^2 - 6b $

Let's solve problem 8: $ 8(8n - 2) $

# Explanation:
## Step 1: Apply the distributive property
$ 8\times8n - 8\times2 $ (using $ a(b - c)=ab - ac $ with $ a = 8 $, $ b = 8n $, $ c = 2 $)

## Step 2: Simplify each product
First product: $ 8\times8n = 64n $
Second product: $ 8\times2 = 16 $

So the result is $ 64n - 16 $

# Answer:
$ 64n - 16 $

Let's solve problem 9: $ 3(8p + 5) $

# Explanation:
## Step 1: Apply the distributive property
Using $ a(b + c)=ab + ac $ with $ a = 3 $, $ b = 8p $, $ c = 5 $
$ 3\times8p + 3\times5 $

## Step 2: Simplify each product
First product: $ 3\times8p = 24p $
Second product: $ 3\times5 = 15 $

So we get $ 24p + 15 $

# Answer:
$ 24p + 15 $

Let's solve problem 10: $ 4b(4b - 6) $

# Explanation:
## Step 1: Apply the distributive property
$ 4b\times4b - 4b\times6 $ (using $ a(b - c)=ab - ac $ with $ a = 4b $, $ b = 4b $, $ c = 6 $)

## Step 2: Simplify each product
First product: $ 4b\times4b = 16b^2 $ ( $ 4\times4 = 16 $ and $ b\times b = b^2 $)
Second product: $ 4b\times6 = 24b $

So the result is $ 16b^2 - 24b $

# Answer:
$ 16b^2 - 24b $

Let's solve problem 11: $ 3(3n - 4) $

# Explanation:
## Step 1: Apply the distributive property
$ 3\times3n - 3\times4 $ (using $ a(b - c)=ab - ac $ with $ a = 3 $, $ b = 3n $, $ c = 4 $)

## Step 2: Simplify each product
First product: $ 3\times3n = 9n $
Second product: $ 3\times4 = 12 $

So we have $ 9n - 12 $

# Answer:
$ 9n - 12 $

Let's solve problem 12: $ 2(6x - 7) $

# Explanation:
## Step 1: Apply the distributive property
$ 2\times6x - 2\times7 $ (using $ a(b - c)=ab - ac $ with $ a = 2 $, $ b = 6x $, $ c = 7 $)

## Step 2: Simplify each product
First product: $ 2\times6x = 12x $
Second product: $ 2\times7 = 14 $

So the result is $ 12x - 14 $

# Answer:
$ 12x - 14 $

Let's solve problem 13: $ 3(5n - 2) $

# Explanation:
## Step 1: Apply the distributive property
$ 3\times5n - 3\times2 $ (using $ a(b - c)=ab - ac $ with $ a = 3 $, $ b = 5n $, $ c = 2 $)

## Step 2: Simplify each product
First product: $ 3\times5n = 15n $
Second product: $ 3\times2 = 6 $

So we get $ 15n - 6 $

# Answer:
$ 15n - 6 $

Let's solve problem 14: $ 5(3x - 1) $

# Explanation:
## Step 1: Apply the distributive property
$ 5\times3x - 5\times1 $ (using $ a(b - c)=ab - ac $ with $ a = 5 $, $ b = 3x $, $ c = 1 $)

## Step 2: Simplify each product
First product: $ 5\times3x = 15x $
Second product: $ 5\times1 = 5 $

So the result is $ 15x - 5 $

# Answer:
$ 15x - 5 $

Let's solve problem 15: $ 6x^2(3x - 6) $

# Explanation:
## Step 1: Apply the distributive property
$ 6x^2\times3x - 6x^2\times6 $ (using $ a(b - c)=ab - ac $ with $ a = 6x^2 $, $ b = 3x $, $ c = 6 $)

## Step 2: Simplify each product
First product: $ 6x^2\times3x = 18x^{3} $ (using $ a^m\times a^n=a^{m + n} $, $ x^2\times x=x^{2 + 1}=x^3 $ and $ 6\times3 = 18 $)
Second product: $ 6x^2\times6 = 36x^2 $

So we have $ 18x^3 - 36x^2 $

# Answer:
$ 18x^3 - 36x^2 $

Let's solve problem 16: $ 7(8x + 2) $

# Explanation:
## Step 1: Apply the distributive property
Using $ a(b + c)=ab + ac $ with $ a = 7 $, $ b = 8x $, $ c = 2 $
$ 7\times8x + 7\times2 $

## Step 2: Simplify each product
First product: $ 7\times8x = 56x $
Second product: $ 7\times2 = 14 $

So the result is $ 56x + 14 $

# Answer:
$ 56x + 14 $

Let's solve problem 17: $ 4r(r + 1) $

# Explanation:
## Step 1: Apply the distributive property
$ 4r\times r + 4r\times1 $ (using $ a(b + c)=ab + ac $ with $ a = 4r $, $ b = r $, $ c = 1 $)

## Step 2: Simplify each product
First product: $ 4r\times r = 4r^2 $ ( $ r\times r = r^2 $ and $ 4\times1 = 4 $)
Second product: $ 4r\times1 = 4r $

So we get $ 4r^2 + 4r $

# Answer:
$ 4r^2 + 4r $

Let's solve problem 18: $ 2x^2(5x + 3) $

# Explanation:
## Step 1: Apply the distributive property
$ 2x^2\times5x + 2x^2\times3 $ (using $ a(b + c)=ab + ac $ with $ a = 2x^2 $, $ b = 5x $, $ c = 3 $)

## Step 2: Simplify each product
First product: $ 2x^2\times5x = 10x^3 $ ( $ 2\times5 = 10 $ and $ x^2\times x=x^{2 + 1}=x^3 $)
Second product: $ 2x^2\times3 = 6x^2 $

So the result is $ 10x^3 + 6x^2 $

# Answer:
$ 10x^3 + 6x^2 $

Let's solve problem 19: $ 5p(6p - 6) $

# Explanation:
## Step 1: Apply the distributive property
$ 5p\times6p - 5p\times6 $ (using $ a(b - c)=ab - ac $ with $ a = 5p $, $ b = 6p $, $ c = 6 $)

## Step 2: Simplify each product
First product: $ 5p\times6p = 30p^2 $ ( $ 5\times6 = 30 $ and $ p\times p = p^2 $)
Second product: $ 5p\times6 = 30p $

So we have $ 30p^2 - 30p $

# Answer:
$ 30p^2 - 30p $

Let's solve problem 20: $ 3v(3v + 4) $

# Explanation:
## Step 1: Apply the distributive property
$ 3v\times3v + 3v\times4 $ (using $ a(b + c)=ab + ac $ with $ a = 3v $, $ b = 3v $, $ c = 4 $)

## Step 2: Simplify each product
First product: $ 3v\times3v = 9v^2 $ ( $ 3\times3 = 9 $ and $ v\times v = v^2 $)
Second product: $ 3v\times4 = 12v $

So the result is $ 9v^2 + 12v $

# Answer:
$ 9v^2 + 12v $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step 1: Apply the distributive property (multiplication over subtraction)

Step 2: Simplify each product

Step 1: Apply the distributive property

Step 2: Simplify each product

Step 1: Apply the distributive property

Step 2: Simplify each product

Answer: