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Step-by-step solution for: 9 Best Multiplying Polynomials Worksheet For Students - The Teach ...
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Step-by-step solution for: 9 Best Multiplying Polynomials Worksheet For Students - The Teach ...
To solve the problem of finding each product of the given polynomial expressions, we will use the distributive property (also known as the FOIL method for binomials) and expand each expression step by step. Let's go through each problem systematically.
---
Using the distributive property:
\[
(2r + 3)(7r + 7) = 2r \cdot 7r + 2r \cdot 7 + 3 \cdot 7r + 3 \cdot 7
\]
\[
= 14r^2 + 14r + 21r + 21
\]
\[
= 14r^2 + 35r + 21
\]
Answer:
\[
\boxed{14r^2 + 35r + 21}
\]
---
Using the distributive property:
\[
(8b + 6)(b - 5) = 8b \cdot b + 8b \cdot (-5) + 6 \cdot b + 6 \cdot (-5)
\]
\[
= 8b^2 - 40b + 6b - 30
\]
\[
= 8b^2 - 34b - 30
\]
Answer:
\[
\boxed{8b^2 - 34b - 30}
\]
---
Using the distributive property:
\[
(7n + 5)(4n + 6) = 7n \cdot 4n + 7n \cdot 6 + 5 \cdot 4n + 5 \cdot 6
\]
\[
= 28n^2 + 42n + 20n + 30
\]
\[
= 28n^2 + 62n + 30
\]
Answer:
\[
\boxed{28n^2 + 62n + 30}
\]
---
Using the distributive property:
\[
(5x + 8)(6x + 3) = 5x \cdot 6x + 5x \cdot 3 + 8 \cdot 6x + 8 \cdot 3
\]
\[
= 30x^2 + 15x + 48x + 24
\]
\[
= 30x^2 + 63x + 24
\]
Answer:
\[
\boxed{30x^2 + 63x + 24}
\]
---
Using the distributive property:
\[
(a - 7)(2a + 2) = a \cdot 2a + a \cdot 2 + (-7) \cdot 2a + (-7) \cdot 2
\]
\[
= 2a^2 + 2a - 14a - 14
\]
\[
= 2a^2 - 12a - 14
\]
Answer:
\[
\boxed{2a^2 - 12a - 14}
\]
---
Using the distributive property:
\[
(4v + 8)(8v^2 - 3v - 4) = 4v \cdot 8v^2 + 4v \cdot (-3v) + 4v \cdot (-4) + 8 \cdot 8v^2 + 8 \cdot (-3v) + 8 \cdot (-4)
\]
\[
= 32v^3 - 12v^2 - 16v + 64v^2 - 24v - 32
\]
Combine like terms:
\[
= 32v^3 + (64v^2 - 12v^2) + (-16v - 24v) - 32
\]
\[
= 32v^3 + 52v^2 - 40v - 32
\]
Answer:
\[
\boxed{32v^3 + 52v^2 - 40v - 32}
\]
---
Using the distributive property:
\[
(x - 7)(5x^2 - 3x - 5) = x \cdot 5x^2 + x \cdot (-3x) + x \cdot (-5) + (-7) \cdot 5x^2 + (-7) \cdot (-3x) + (-7) \cdot (-5)
\]
\[
= 5x^3 - 3x^2 - 5x - 35x^2 + 21x + 35
\]
Combine like terms:
\[
= 5x^3 + (-3x^2 - 35x^2) + (-5x + 21x) + 35
\]
\[
= 5x^3 - 38x^2 + 16x + 35
\]
Answer:
\[
\boxed{5x^3 - 38x^2 + 16x + 35}
\]
---
Using the distributive property:
\[
(6x - 4)(7x^2 + x - 4) = 6x \cdot 7x^2 + 6x \cdot x + 6x \cdot (-4) + (-4) \cdot 7x^2 + (-4) \cdot x + (-4) \cdot (-4)
\]
\[
= 42x^3 + 6x^2 - 24x - 28x^2 - 4x + 16
\]
Combine like terms:
\[
= 42x^3 + (6x^2 - 28x^2) + (-24x - 4x) + 16
\]
\[
= 42x^3 - 22x^2 - 28x + 16
\]
Answer:
\[
\boxed{42x^3 - 22x^2 - 28x + 16}
\]
---
Using the distributive property:
\[
(3n - 2)(3n^2 - 8n - 5) = 3n \cdot 3n^2 + 3n \cdot (-8n) + 3n \cdot (-5) + (-2) \cdot 3n^2 + (-2) \cdot (-8n) + (-2) \cdot (-5)
\]
\[
= 9n^3 - 24n^2 - 15n - 6n^2 + 16n + 10
\]
Combine like terms:
\[
= 9n^3 + (-24n^2 - 6n^2) + (-15n + 16n) + 10
\]
\[
= 9n^3 - 30n^2 + n + 10
\]
Answer:
\[
\boxed{9n^3 - 30n^2 + n + 10}
\]
---
Using the distributive property:
\[
(5k - 5)(k^2 - 4k - 5) = 5k \cdot k^2 + 5k \cdot (-4k) + 5k \cdot (-5) + (-5) \cdot k^2 + (-5) \cdot (-4k) + (-5) \cdot (-5)
\]
\[
= 5k^3 - 20k^2 - 25k - 5k^2 + 20k + 25
\]
Combine like terms:
\[
= 5k^3 + (-20k^2 - 5k^2) + (-25k + 20k) + 25
\]
\[
= 5k^3 - 25k^2 - 5k + 25
\]
Answer:
\[
\boxed{5k^3 - 25k^2 - 5k + 25}
\]
---
Using the distributive property:
\[
(2x^2 + 6x - 8)(2x^2 - 6x - 3) = 2x^2 \cdot 2x^2 + 2x^2 \cdot (-6x) + 2x^2 \cdot (-3) + 6x \cdot 2x^2 + 6x \cdot (-6x) + 6x \cdot (-3) + (-8) \cdot 2x^2 + (-8) \cdot (-6x) + (-8) \cdot (-3)
\]
\[
= 4x^4 - 12x^3 - 6x^2 + 12x^3 - 36x^2 - 18x - 16x^2 + 48x + 24
\]
Combine like terms:
\[
= 4x^4 + (-12x^3 + 12x^3) + (-6x^2 - 36x^2 - 16x^2) + (-18x + 48x) + 24
\]
\[
= 4x^4 - 58x^2 + 30x + 24
\]
Answer:
\[
\boxed{4x^4 - 58x^2 + 30x + 24}
\]
---
Using the distributive property:
\[
(2p^2 - 7p + 8)(2p^2 - 5p - 3) = 2p^2 \cdot 2p^2 + 2p^2 \cdot (-5p) + 2p^2 \cdot (-3) + (-7p) \cdot 2p^2 + (-7p) \cdot (-5p) + (-7p) \cdot (-3) + 8 \cdot 2p^2 + 8 \cdot (-5p) + 8 \cdot (-3)
\]
\[
= 4p^4 - 10p^3 - 6p^2 - 14p^3 + 35p^2 + 21p + 16p^2 - 40p - 24
\]
Combine like terms:
\[
= 4p^4 + (-10p^3 - 14p^3) + (-6p^2 + 35p^2 + 16p^2) + (21p - 40p) - 24
\]
\[
= 4p^4 - 24p^3 + 45p^2 - 19p - 24
\]
Answer:
\[
\boxed{4p^4 - 24p^3 + 45p^2 - 19p - 24}
\]
---
Using the distributive property:
\[
(5n^2 + 3n - 8)(4n^2 - 6n - 2) = 5n^2 \cdot 4n^2 + 5n^2 \cdot (-6n) + 5n^2 \cdot (-2) + 3n \cdot 4n^2 + 3n \cdot (-6n) + 3n \cdot (-2) + (-8) \cdot 4n^2 + (-8) \cdot (-6n) + (-8) \cdot (-2)
\]
\[
= 20n^4 - 30n^3 - 10n^2 + 12n^3 - 18n^2 - 6n - 32n^2 + 48n + 16
\]
Combine like terms:
\[
= 20n^4 + (-30n^3 + 12n^3) + (-10n^2 - 18n^2 - 32n^2) + (-6n + 48n) + 16
\]
\[
= 20n^4 - 18n^3 - 60n^2 + 42n + 16
\]
Answer:
\[
\boxed{20n^4 - 18n^3 - 60n^2 + 42n + 16}
\]
---
Using the distributive property:
\[
(8m^2 + 8m + 3)(2m^2 + 8m + 4) = 8m^2 \cdot 2m^2 + 8m^2 \cdot 8m + 8m^2 \cdot 4 + 8m \cdot 2m^2 + 8m \cdot 8m + 8m \cdot 4 + 3 \cdot 2m^2 + 3 \cdot 8m + 3 \cdot 4
\]
\[
= 16m^4 + 64m^3 + 32m^2 + 16m^3 + 64m^2 + 32m + 6m^2 + 24m + 12
\]
Combine like terms:
\[
= 16m^4 + (64m^3 + 16m^3) + (32m^2 + 64m^2 + 6m^2) + (32m + 24m) + 12
\]
\[
= 16m^4 + 80m^3 + 102m^2 + 56m + 12
\]
Answer:
\[
\boxed{16m^4 + 80m^3 + 102m^2 + 56m + 12}
\]
---
Using the distributive property:
\[
(4x^2 - 7x - 1)(6x^2 + x - 7) = 4x^2 \cdot 6x^2 + 4x^2 \cdot x + 4x^2 \cdot (-7) + (-7x) \cdot 6x^2 + (-7x) \cdot x + (-7x) \cdot (-7) + (-1) \cdot 6x^2 + (-1) \cdot x + (-1) \cdot (-7)
\]
\[
= 24x^4 + 4x^3 - 28x^2 - 42x^3 - 7x^2 + 49x - 6x^2 - x + 7
\]
Combine like terms:
\[
= 24x^4 + (4x^3 - 42x^3) + (-28x^2 - 7x^2 - 6x^2) + (49x - x) + 7
\]
\[
= 24x^4 - 38x^3 - 41x^2 + 48x + 7
\]
Answer:
\[
\boxed{24x^4 - 38x^3 - 41x^2 + 48x + 7}
\]
---
Using the distributive property:
\[
(2r^2 + 5r - 1)(5r^2 + 2r - 8) = 2r^2 \cdot 5r^2 + 2r^2 \cdot 2r + 2r^2 \cdot (-8) + 5r \cdot 5r^2 + 5r \cdot 2r + 5r \cdot (-8) + (-1) \cdot 5r^2 + (-1) \cdot 2r + (-1) \cdot (-8)
\]
\[
= 10r^4 + 4r^3 - 16r^2 + 25r^3 + 10r^2 - 40r - 5r^2 - 2r + 8
\]
Combine like terms:
\[
= 10r^4 + (4r^3 + 25r^3) + (-16r^2 + 10r^2 - 5r^2) + (-40r - 2r) + 8
\]
\[
= 10r^4 + 29r^3 - 11r^2 - 42r + 8
\]
Answer:
\[
\boxed{10r^4 + 29r^3 - 11r^2 - 42r + 8}
\]
---
This is a square of a trinomial. Use the formula \((A + B + C)^2 = A^2 + B^2 + C^2 + 2AB + 2AC + 2BC\):
\[
(b^2 - 4b + 6)^2 = (b^2)^2 + (-4b)^2 + 6^2 + 2(b^2)(-4b) + 2(b^2)(6) + 2(-4b)(6)
\]
\[
= b^4 + 16b^2 + 36 - 8b^3 + 12b^2 - 48b
\]
Combine like terms:
\[
= b^4 - 8b^3 + (16b^2 + 12b^2) - 48b + 36
\]
\[
= b^4 - 8b^3 + 28b^2 - 48b + 36
\]
Answer:
\[
\boxed{b^4 - 8b^3 + 28b^2 - 48b + 36}
\]
---
Using the distributive property:
\[
(7n^2 + 8n + 7)(7n^2 + n - 5) = 7n^2 \cdot 7n^2 + 7n^2 \cdot n + 7n^2 \cdot (-5) + 8n \cdot 7n^2 + 8n \cdot n + 8n \cdot (-5) + 7 \cdot 7n^2 + 7 \cdot n + 7 \cdot (-5)
\]
\[
= 49n^4 + 7n^3 - 35n^2 + 56n^3 + 8n^2 - 40n + 49n^2 + 7n - 35
\]
Combine like terms:
\[
= 49n^4 + (7n^3 + 56n^3) + (-35n^2 + 8n^2 + 49n^2) + (-40n + 7n) - 35
\]
\[
= 49n^4 + 63n^3 + 22n^2 - 33n - 35
\]
Answer:
\[
\boxed{49n^4 + 63n^3 + 22n^2 - 33n - 35}
\]
---
\[
\boxed{
\begin{aligned}
1. & \ 14r^2 + 35r + 21 \\
2. & \ 8b^2 - 34b - 30 \\
3. & \ 28n^2 + 62n + 30 \\
4. & \ 30x^2 + 63x + 24 \\
5. & \ 2a^2 - 12a - 14 \\
6. & \ 32v^3 + 52v^2 - 40v - 32 \\
7. & \ 5x^3 - 38x^2 + 16x + 35 \\
8. & \ 42x^3 - 22x^2 - 28x + 16 \\
9. & \ 9n^3 - 30n^2 + n + 10 \\
10. & \ 5k^3 - 25k^2 - 5k + 25 \\
11. & \ 4x^4 - 58x^2 + 30x + 24 \\
12. & \ 4p^4 - 24p^3 + 45p^2 - 19p - 24 \\
13. & \ 20n^4 - 18n^3 - 60n^2 + 42n + 16 \\
14. & \ 16m^4 + 80m^3 + 102m^2 + 56m + 12 \\
15. & \ 24x^4 - 38x^3 - 41x^2 + 48x + 7 \\
16. & \ 10r^4 + 29r^3 - 11r^2 - 42r + 8 \\
17. & \ b^4 - 8b^3 + 28b^2 - 48b + 36 \\
18. & \ 49n^4 + 63n^3 + 22n^2 - 33n - 35 \\
\end{aligned}
}
\]
---
1. \( (2r + 3)(7r + 7) \)
Using the distributive property:
\[
(2r + 3)(7r + 7) = 2r \cdot 7r + 2r \cdot 7 + 3 \cdot 7r + 3 \cdot 7
\]
\[
= 14r^2 + 14r + 21r + 21
\]
\[
= 14r^2 + 35r + 21
\]
Answer:
\[
\boxed{14r^2 + 35r + 21}
\]
---
2. \( (8b + 6)(b - 5) \)
Using the distributive property:
\[
(8b + 6)(b - 5) = 8b \cdot b + 8b \cdot (-5) + 6 \cdot b + 6 \cdot (-5)
\]
\[
= 8b^2 - 40b + 6b - 30
\]
\[
= 8b^2 - 34b - 30
\]
Answer:
\[
\boxed{8b^2 - 34b - 30}
\]
---
3. \( (7n + 5)(4n + 6) \)
Using the distributive property:
\[
(7n + 5)(4n + 6) = 7n \cdot 4n + 7n \cdot 6 + 5 \cdot 4n + 5 \cdot 6
\]
\[
= 28n^2 + 42n + 20n + 30
\]
\[
= 28n^2 + 62n + 30
\]
Answer:
\[
\boxed{28n^2 + 62n + 30}
\]
---
4. \( (5x + 8)(6x + 3) \)
Using the distributive property:
\[
(5x + 8)(6x + 3) = 5x \cdot 6x + 5x \cdot 3 + 8 \cdot 6x + 8 \cdot 3
\]
\[
= 30x^2 + 15x + 48x + 24
\]
\[
= 30x^2 + 63x + 24
\]
Answer:
\[
\boxed{30x^2 + 63x + 24}
\]
---
5. \( (a - 7)(2a + 2) \)
Using the distributive property:
\[
(a - 7)(2a + 2) = a \cdot 2a + a \cdot 2 + (-7) \cdot 2a + (-7) \cdot 2
\]
\[
= 2a^2 + 2a - 14a - 14
\]
\[
= 2a^2 - 12a - 14
\]
Answer:
\[
\boxed{2a^2 - 12a - 14}
\]
---
6. \( (4v + 8)(8v^2 - 3v - 4) \)
Using the distributive property:
\[
(4v + 8)(8v^2 - 3v - 4) = 4v \cdot 8v^2 + 4v \cdot (-3v) + 4v \cdot (-4) + 8 \cdot 8v^2 + 8 \cdot (-3v) + 8 \cdot (-4)
\]
\[
= 32v^3 - 12v^2 - 16v + 64v^2 - 24v - 32
\]
Combine like terms:
\[
= 32v^3 + (64v^2 - 12v^2) + (-16v - 24v) - 32
\]
\[
= 32v^3 + 52v^2 - 40v - 32
\]
Answer:
\[
\boxed{32v^3 + 52v^2 - 40v - 32}
\]
---
7. \( (x - 7)(5x^2 - 3x - 5) \)
Using the distributive property:
\[
(x - 7)(5x^2 - 3x - 5) = x \cdot 5x^2 + x \cdot (-3x) + x \cdot (-5) + (-7) \cdot 5x^2 + (-7) \cdot (-3x) + (-7) \cdot (-5)
\]
\[
= 5x^3 - 3x^2 - 5x - 35x^2 + 21x + 35
\]
Combine like terms:
\[
= 5x^3 + (-3x^2 - 35x^2) + (-5x + 21x) + 35
\]
\[
= 5x^3 - 38x^2 + 16x + 35
\]
Answer:
\[
\boxed{5x^3 - 38x^2 + 16x + 35}
\]
---
8. \( (6x - 4)(7x^2 + x - 4) \)
Using the distributive property:
\[
(6x - 4)(7x^2 + x - 4) = 6x \cdot 7x^2 + 6x \cdot x + 6x \cdot (-4) + (-4) \cdot 7x^2 + (-4) \cdot x + (-4) \cdot (-4)
\]
\[
= 42x^3 + 6x^2 - 24x - 28x^2 - 4x + 16
\]
Combine like terms:
\[
= 42x^3 + (6x^2 - 28x^2) + (-24x - 4x) + 16
\]
\[
= 42x^3 - 22x^2 - 28x + 16
\]
Answer:
\[
\boxed{42x^3 - 22x^2 - 28x + 16}
\]
---
9. \( (3n - 2)(3n^2 - 8n - 5) \)
Using the distributive property:
\[
(3n - 2)(3n^2 - 8n - 5) = 3n \cdot 3n^2 + 3n \cdot (-8n) + 3n \cdot (-5) + (-2) \cdot 3n^2 + (-2) \cdot (-8n) + (-2) \cdot (-5)
\]
\[
= 9n^3 - 24n^2 - 15n - 6n^2 + 16n + 10
\]
Combine like terms:
\[
= 9n^3 + (-24n^2 - 6n^2) + (-15n + 16n) + 10
\]
\[
= 9n^3 - 30n^2 + n + 10
\]
Answer:
\[
\boxed{9n^3 - 30n^2 + n + 10}
\]
---
10. \( (5k - 5)(k^2 - 4k - 5) \)
Using the distributive property:
\[
(5k - 5)(k^2 - 4k - 5) = 5k \cdot k^2 + 5k \cdot (-4k) + 5k \cdot (-5) + (-5) \cdot k^2 + (-5) \cdot (-4k) + (-5) \cdot (-5)
\]
\[
= 5k^3 - 20k^2 - 25k - 5k^2 + 20k + 25
\]
Combine like terms:
\[
= 5k^3 + (-20k^2 - 5k^2) + (-25k + 20k) + 25
\]
\[
= 5k^3 - 25k^2 - 5k + 25
\]
Answer:
\[
\boxed{5k^3 - 25k^2 - 5k + 25}
\]
---
11. \( (2x^2 + 6x - 8)(2x^2 - 6x - 3) \)
Using the distributive property:
\[
(2x^2 + 6x - 8)(2x^2 - 6x - 3) = 2x^2 \cdot 2x^2 + 2x^2 \cdot (-6x) + 2x^2 \cdot (-3) + 6x \cdot 2x^2 + 6x \cdot (-6x) + 6x \cdot (-3) + (-8) \cdot 2x^2 + (-8) \cdot (-6x) + (-8) \cdot (-3)
\]
\[
= 4x^4 - 12x^3 - 6x^2 + 12x^3 - 36x^2 - 18x - 16x^2 + 48x + 24
\]
Combine like terms:
\[
= 4x^4 + (-12x^3 + 12x^3) + (-6x^2 - 36x^2 - 16x^2) + (-18x + 48x) + 24
\]
\[
= 4x^4 - 58x^2 + 30x + 24
\]
Answer:
\[
\boxed{4x^4 - 58x^2 + 30x + 24}
\]
---
12. \( (2p^2 - 7p + 8)(2p^2 - 5p - 3) \)
Using the distributive property:
\[
(2p^2 - 7p + 8)(2p^2 - 5p - 3) = 2p^2 \cdot 2p^2 + 2p^2 \cdot (-5p) + 2p^2 \cdot (-3) + (-7p) \cdot 2p^2 + (-7p) \cdot (-5p) + (-7p) \cdot (-3) + 8 \cdot 2p^2 + 8 \cdot (-5p) + 8 \cdot (-3)
\]
\[
= 4p^4 - 10p^3 - 6p^2 - 14p^3 + 35p^2 + 21p + 16p^2 - 40p - 24
\]
Combine like terms:
\[
= 4p^4 + (-10p^3 - 14p^3) + (-6p^2 + 35p^2 + 16p^2) + (21p - 40p) - 24
\]
\[
= 4p^4 - 24p^3 + 45p^2 - 19p - 24
\]
Answer:
\[
\boxed{4p^4 - 24p^3 + 45p^2 - 19p - 24}
\]
---
13. \( (5n^2 + 3n - 8)(4n^2 - 6n - 2) \)
Using the distributive property:
\[
(5n^2 + 3n - 8)(4n^2 - 6n - 2) = 5n^2 \cdot 4n^2 + 5n^2 \cdot (-6n) + 5n^2 \cdot (-2) + 3n \cdot 4n^2 + 3n \cdot (-6n) + 3n \cdot (-2) + (-8) \cdot 4n^2 + (-8) \cdot (-6n) + (-8) \cdot (-2)
\]
\[
= 20n^4 - 30n^3 - 10n^2 + 12n^3 - 18n^2 - 6n - 32n^2 + 48n + 16
\]
Combine like terms:
\[
= 20n^4 + (-30n^3 + 12n^3) + (-10n^2 - 18n^2 - 32n^2) + (-6n + 48n) + 16
\]
\[
= 20n^4 - 18n^3 - 60n^2 + 42n + 16
\]
Answer:
\[
\boxed{20n^4 - 18n^3 - 60n^2 + 42n + 16}
\]
---
14. \( (8m^2 + 8m + 3)(2m^2 + 8m + 4) \)
Using the distributive property:
\[
(8m^2 + 8m + 3)(2m^2 + 8m + 4) = 8m^2 \cdot 2m^2 + 8m^2 \cdot 8m + 8m^2 \cdot 4 + 8m \cdot 2m^2 + 8m \cdot 8m + 8m \cdot 4 + 3 \cdot 2m^2 + 3 \cdot 8m + 3 \cdot 4
\]
\[
= 16m^4 + 64m^3 + 32m^2 + 16m^3 + 64m^2 + 32m + 6m^2 + 24m + 12
\]
Combine like terms:
\[
= 16m^4 + (64m^3 + 16m^3) + (32m^2 + 64m^2 + 6m^2) + (32m + 24m) + 12
\]
\[
= 16m^4 + 80m^3 + 102m^2 + 56m + 12
\]
Answer:
\[
\boxed{16m^4 + 80m^3 + 102m^2 + 56m + 12}
\]
---
15. \( (4x^2 - 7x - 1)(6x^2 + x - 7) \)
Using the distributive property:
\[
(4x^2 - 7x - 1)(6x^2 + x - 7) = 4x^2 \cdot 6x^2 + 4x^2 \cdot x + 4x^2 \cdot (-7) + (-7x) \cdot 6x^2 + (-7x) \cdot x + (-7x) \cdot (-7) + (-1) \cdot 6x^2 + (-1) \cdot x + (-1) \cdot (-7)
\]
\[
= 24x^4 + 4x^3 - 28x^2 - 42x^3 - 7x^2 + 49x - 6x^2 - x + 7
\]
Combine like terms:
\[
= 24x^4 + (4x^3 - 42x^3) + (-28x^2 - 7x^2 - 6x^2) + (49x - x) + 7
\]
\[
= 24x^4 - 38x^3 - 41x^2 + 48x + 7
\]
Answer:
\[
\boxed{24x^4 - 38x^3 - 41x^2 + 48x + 7}
\]
---
16. \( (2r^2 + 5r - 1)(5r^2 + 2r - 8) \)
Using the distributive property:
\[
(2r^2 + 5r - 1)(5r^2 + 2r - 8) = 2r^2 \cdot 5r^2 + 2r^2 \cdot 2r + 2r^2 \cdot (-8) + 5r \cdot 5r^2 + 5r \cdot 2r + 5r \cdot (-8) + (-1) \cdot 5r^2 + (-1) \cdot 2r + (-1) \cdot (-8)
\]
\[
= 10r^4 + 4r^3 - 16r^2 + 25r^3 + 10r^2 - 40r - 5r^2 - 2r + 8
\]
Combine like terms:
\[
= 10r^4 + (4r^3 + 25r^3) + (-16r^2 + 10r^2 - 5r^2) + (-40r - 2r) + 8
\]
\[
= 10r^4 + 29r^3 - 11r^2 - 42r + 8
\]
Answer:
\[
\boxed{10r^4 + 29r^3 - 11r^2 - 42r + 8}
\]
---
17. \( (b^2 - 4b + 6)^2 \)
This is a square of a trinomial. Use the formula \((A + B + C)^2 = A^2 + B^2 + C^2 + 2AB + 2AC + 2BC\):
\[
(b^2 - 4b + 6)^2 = (b^2)^2 + (-4b)^2 + 6^2 + 2(b^2)(-4b) + 2(b^2)(6) + 2(-4b)(6)
\]
\[
= b^4 + 16b^2 + 36 - 8b^3 + 12b^2 - 48b
\]
Combine like terms:
\[
= b^4 - 8b^3 + (16b^2 + 12b^2) - 48b + 36
\]
\[
= b^4 - 8b^3 + 28b^2 - 48b + 36
\]
Answer:
\[
\boxed{b^4 - 8b^3 + 28b^2 - 48b + 36}
\]
---
18. \( (7n^2 + 8n + 7)(7n^2 + n - 5) \)
Using the distributive property:
\[
(7n^2 + 8n + 7)(7n^2 + n - 5) = 7n^2 \cdot 7n^2 + 7n^2 \cdot n + 7n^2 \cdot (-5) + 8n \cdot 7n^2 + 8n \cdot n + 8n \cdot (-5) + 7 \cdot 7n^2 + 7 \cdot n + 7 \cdot (-5)
\]
\[
= 49n^4 + 7n^3 - 35n^2 + 56n^3 + 8n^2 - 40n + 49n^2 + 7n - 35
\]
Combine like terms:
\[
= 49n^4 + (7n^3 + 56n^3) + (-35n^2 + 8n^2 + 49n^2) + (-40n + 7n) - 35
\]
\[
= 49n^4 + 63n^3 + 22n^2 - 33n - 35
\]
Answer:
\[
\boxed{49n^4 + 63n^3 + 22n^2 - 33n - 35}
\]
---
Final Answers:
\[
\boxed{
\begin{aligned}
1. & \ 14r^2 + 35r + 21 \\
2. & \ 8b^2 - 34b - 30 \\
3. & \ 28n^2 + 62n + 30 \\
4. & \ 30x^2 + 63x + 24 \\
5. & \ 2a^2 - 12a - 14 \\
6. & \ 32v^3 + 52v^2 - 40v - 32 \\
7. & \ 5x^3 - 38x^2 + 16x + 35 \\
8. & \ 42x^3 - 22x^2 - 28x + 16 \\
9. & \ 9n^3 - 30n^2 + n + 10 \\
10. & \ 5k^3 - 25k^2 - 5k + 25 \\
11. & \ 4x^4 - 58x^2 + 30x + 24 \\
12. & \ 4p^4 - 24p^3 + 45p^2 - 19p - 24 \\
13. & \ 20n^4 - 18n^3 - 60n^2 + 42n + 16 \\
14. & \ 16m^4 + 80m^3 + 102m^2 + 56m + 12 \\
15. & \ 24x^4 - 38x^3 - 41x^2 + 48x + 7 \\
16. & \ 10r^4 + 29r^3 - 11r^2 - 42r + 8 \\
17. & \ b^4 - 8b^3 + 28b^2 - 48b + 36 \\
18. & \ 49n^4 + 63n^3 + 22n^2 - 33n - 35 \\
\end{aligned}
}
\]
Parent Tip: Review the logic above to help your child master the concept of multiplication of polynomials worksheet.
