Here is the step-by-step solution to all 10 polynomial operations:
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① (8b - 2)(3b² + 2b - 2)
Use distributive property (FOIL or box method):
= 8b(3b²) + 8b(2b) + 8b(-2) -2(3b²) -2(2b) -2(-2)
= 24b³ + 16b² - 16b - 6b² - 4b + 4
Combine like terms:
=
24b³ + (16b² - 6b²) + (-16b - 4b) + 4
=
24b³ + 10b² - 20b + 4
✔ Final Answer:
24b³ + 10b² - 20b + 4
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② (3x² - 2x + 1) + (-x² + 3x + 1)
Add like terms:
= (3x² - x²) + (-2x + 3x) + (1 + 1)
=
2x² + x + 2
✔ Final Answer:
2x² + x + 2
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③ (-3m² + m) + (4m² + 6m)
Add like terms:
= (-3m² + 4m²) + (m + 6m)
=
m² + 7m
✔ Final Answer:
m² + 7m
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④ (7a² - a + 4) - (3a² - 4a - 3)
Distribute the minus sign:
= 7a² - a + 4 - 3a² + 4a + 3
Combine like terms:
= (7a² - 3a²) + (-a + 4a) + (4 + 3)
=
4a² + 3a + 7
✔ Final Answer:
4a² + 3a + 7
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⑤ (-3x² + 6x³ - 4 - x) ÷ (2x + 1)
First, write dividend in standard form (descending powers):
= (6x³ - 3x² - x - 4) ÷ (2x + 1)
Use
polynomial long division:
Divide 6x³ by 2x →
3x²
Multiply: 3x²(2x + 1) = 6x³ + 3x²
Subtract: (6x³ - 3x²) - (6x³ + 3x²) = -6x²
Bring down next term: -6x² - x
Divide -6x² by 2x →
-3x
Multiply: -3x(2x + 1) = -6x² - 3x
Subtract: (-6x² - x) - (-6x² - 3x) = 2x
Bring down -4: 2x - 4
Divide 2x by 2x →
1
Multiply: 1(2x + 1) = 2x + 1
Subtract: (2x - 4) - (2x + 1) =
-5
So quotient is
3x² - 3x + 1, remainder
-5
✔ Final Answer:
3x² - 3x + 1 - \frac{5}{2x + 1}
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⑥ 2a(5a² + 8a + 8)
Distribute 2a:
= 2a·5a² + 2a·8a + 2a·8
=
10a³ + 16a² + 16a
✔ Final Answer:
10a³ + 16a² + 16a
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⑦ (-18p² + p - 32) - (40 - 13p²)
Distribute the minus:
= -18p² + p - 32 - 40 + 13p²
Combine like terms:
= (-18p² + 13p²) + p + (-32 - 40)
=
-5p² + p - 72
✔ Final Answer:
-5p² + p - 72
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⑧ -8w²y + (4w²y⁴ - w⁴)
No like terms to combine. Just remove parentheses:
=
-8w²y + 4w²y⁴ - w⁴
You can write in standard order (by degree or alphabetically), but no simplification needed.
✔ Final Answer:
4w²y⁴ - 8w²y - w⁴
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⑨ (x - 2)(x² - x + 3)
Use distributive property:
= x(x² - x + 3) - 2(x² - x + 3)
= x³ - x² + 3x - 2x² + 2x - 6
Combine like terms:
= x³ + (-x² - 2x²) + (3x + 2x) - 6
=
x³ - 3x² + 5x - 6
✔ Final Answer:
x³ - 3x² + 5x - 6
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⑩ (5x³ - 13x² - 7) + (16x³ + 8x² - x + 15)
Add like terms:
= (5x³ + 16x³) + (-13x² + 8x²) + (-x) + (-7 + 15)
=
21x³ - 5x² - x + 8
✔ Final Answer:
21x³ - 5x² - x + 8
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##
✔ Final Answers Summary:
1.
24b³ + 10b² - 20b + 4
2.
2x² + x + 2
3.
m² + 7m
4.
4a² + 3a + 7
5.
3x² - 3x + 1 - \frac{5}{2x + 1}
6.
10a³ + 16a² + 16a
7.
-5p² + p - 72
8.
4w²y⁴ - 8w²y - w⁴
9.
x³ - 3x² + 5x - 6
10.
21x³ - 5x² - x + 8
Let me know if you’d like any step explained further!
Parent Tip: Review the logic above to help your child master the concept of polynomial equation worksheet.
