Let’s solve each problem step by step. We’ll combine like terms — that means adding or subtracting terms with the same variable and exponent.
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1) (q² - 8q⁴ - 7) + (3q² + 4 - 5q⁵) + (9q² - 6q - 2)
First, remove parentheses:
= q² - 8q⁴ - 7 + 3q² + 4 - 5q⁵ + 9q² - 6q - 2
Now group like terms:
- q⁵ term:
-5q⁵
- q⁴ term:
-8q⁴
- q² terms: q² + 3q² + 9q² =
13q²
- q term:
-6q
- constants: -7 + 4 - 2 =
-5
Final:
-5q⁵ - 8q⁴ + 13q² - 6q - 5
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2) (6c - 4c⁴ - 3) - (9c + 5c⁴ - 8c²)
Distribute the minus sign to the second group:
= 6c - 4c⁴ - 3 - 9c - 5c⁴ + 8c²
Group like terms:
- c⁴: -4c⁴ - 5c⁴ =
-9c⁴
- c²:
+8c²
- c: 6c - 9c =
-3c
- constant:
-3
Final:
-9c⁴ + 8c² - 3c - 3
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3) (8q⁴ - 3q² + 2q) - (4q² - 5q) + (7q - 9q² + 6q⁴)
Remove parentheses (watch signs!):
= 8q⁴ - 3q² + 2q - 4q² + 5q + 7q - 9q² + 6q⁴
Group like terms:
- q⁴: 8q⁴ + 6q⁴ =
14q⁴
- q²: -3q² - 4q² - 9q² =
-16q²
- q: 2q + 5q + 7q =
14q
Final:
14q⁴ - 16q² + 14q
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4) (6d⁴ - 4) + (9 + 8d³ - 5d⁴)
Remove parentheses:
= 6d⁴ - 4 + 9 + 8d³ - 5d⁴
Group like terms:
- d⁴: 6d⁴ - 5d⁴ =
d⁴
- d³:
+8d³
- constants: -4 + 9 =
5
Final:
d⁴ + 8d³ + 5
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5) (3z² + 5) + (4z³ + 8 - 9z²) + (6z² + 7z - 2)
Remove parentheses:
= 3z² + 5 + 4z³ + 8 - 9z² + 6z² + 7z - 2
Group like terms:
- z³:
+4z³
- z²: 3z² - 9z² + 6z² =
0z² → disappears
- z:
+7z
- constants: 5 + 8 - 2 =
11
Final:
4z³ + 7z + 11
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6) (7p² - 4p⁴ + 3p) - (5p² + 6p³) + (9p³ - 8p⁴ + 2p⁵)
Remove parentheses (distribute minus):
= 7p² - 4p⁴ + 3p - 5p² - 6p³ + 9p³ - 8p⁴ + 2p⁵
Group like terms:
- p⁵:
+2p⁵
- p⁴: -4p⁴ - 8p⁴ =
-12p⁴
- p³: -6p³ + 9p³ =
+3p³
- p²: 7p² - 5p² =
+2p²
- p:
+3p
Final:
2p⁵ - 12p⁴ + 3p³ + 2p² + 3p
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7) (8y² - 2) - (9y⁵ + 5 + 3y²)
Distribute minus:
= 8y² - 2 - 9y⁵ - 5 - 3y²
Group:
- y⁵:
-9y⁵
- y²: 8y² - 3y² =
5y²
- constants: -2 - 5 =
-7
Final:
-9y⁵ + 5y² - 7
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8) (7z² + 5z³ - 4z⁴) - (8z⁴ + 6) - (9z³ + 3z²)
Distribute minuses:
= 7z² + 5z³ - 4z⁴ - 8z⁴ - 6 - 9z³ - 3z²
Group:
- z⁴: -4z⁴ - 8z⁴ =
-12z⁴
- z³: 5z³ - 9z³ =
-4z³
- z²: 7z² - 3z² =
4z²
- constant:
-6
Final:
-12z⁴ - 4z³ + 4z² - 6
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9) (2p - 8p⁵) - (4p + 6p⁴ - 5)
Distribute minus:
= 2p - 8p⁵ - 4p - 6p⁴ + 5
Group:
- p⁵:
-8p⁵
- p⁴:
-6p⁴
- p: 2p - 4p =
-2p
- constant:
+5
Final:
-8p⁵ - 6p⁴ - 2p + 5
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10) (6x⁴ - 5x²) - (2x² - 4x + 9x⁴)
Distribute minus:
= 6x⁴ - 5x² - 2x² + 4x - 9x⁴
Group:
- x⁴: 6x⁴ - 9x⁴ =
-3x⁴
- x²: -5x² - 2x² =
-7x²
- x:
+4x
Final:
-3x⁴ - 7x² + 4x
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Final Answer:
1) -5q⁵ - 8q⁴ + 13q² - 6q - 5
2) -9c⁴ + 8c² - 3c - 3
3) 14q⁴ - 16q² + 14q
4) d⁴ + 8d³ + 5
5) 4z³ + 7z + 11
6) 2p⁵ - 12p⁴ + 3p³ + 2p² + 3p
7) -9y⁵ + 5y² - 7
8) -12z⁴ - 4z³ + 4z² - 6
9) -8p⁵ - 6p⁴ - 2p + 5
10) -3x⁴ - 7x² + 4x
Parent Tip: Review the logic above to help your child master the concept of algebra 1 adding and subtracting polynomials worksheet.
