Adding and Subtracting Polynomials Worksheet for 9th Grade ... - Free Printable

Let’s solve each problem step by step. We’re adding or subtracting polynomials — that means combining like terms (terms with the same variable and exponent).

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Problem 1:
(3a² + 5a – 7) + (–4a² – 8a + 6)

Step 1: Remove parentheses (since it’s addition, signs stay the same):
= 3a² + 5a – 7 – 4a² – 8a + 6

Step 2: Group like terms:
a² terms: 3a² – 4a² = –a²
a terms: 5a – 8a = –3a
constants: –7 + 6 = –1

Final answer for #1: –a² – 3a – 1

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Problem 2:
(3b³ – 5b² + b – 9) + (–b³ + 7b – 2)

Remove parentheses:
= 3b³ – 5b² + b – 9 – b³ + 7b – 2

Group like terms:
b³: 3b³ – b³ = 2b³
b²: –5b² (only one)
b: b + 7b = 8b
constants: –9 – 2 = –11

Final answer for #2: 2b³ – 5b² + 8b – 11

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Problem 3:
(2p² – 6p + 8) – (3p² + 4p – 2)

Subtracting means we change the sign of every term in the second polynomial:

= 2p² – 6p + 8 – 3p² – 4p + 2

Group like terms:
p²: 2p² – 3p² = –p²
p: –6p – 4p = –10p
constants: 8 + 2 = 10

Final answer for #3: –p² – 10p + 10

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Problem 4:
(½x² + ⅓x – ¼) – (¾x² – ⅙x + ½)

Change signs of second polynomial:
= ½x² + ⅓x – ¼ – ¾x² + ⅙x – ½

Group like terms:

x² terms: ½ – ¾ = (2/4 – 3/4) = –¼ → –¼x²
x terms: ⅓ + ⅙ = (2/6 + 1/6) = 3/6 = ½ → ½x
constants: –¼ – ½ = –¼ – 2/4 = –¾

Final answer for #4: –¼x² + ½x – ¾

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Problem 5:
(7c³ – 2c + 9) – (–c³ + 5c – 1)

Change signs of second polynomial:
= 7c³ – 2c + 9 + c³ – 5c + 1

Group like terms:
c³: 7c³ + c³ = 8c³
c: –2c – 5c = –7c
constants: 9 + 1 = 10

Final answer for #5: 8c³ – 7c + 10

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Problem 6:
(4d⁴ – d² + 3) – (2d⁴ – 3d² + 5)

Change signs:
= 4d⁴ – d² + 3 – 2d⁴ + 3d² – 5

Group like terms:
d⁴: 4d⁴ – 2d⁴ = 2d⁴
d²: –d² + 3d² = 2d²
constants: 3 – 5 = –2

Final answer for #6: 2d⁴ + 2d² – 2

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Problem 7:
(5e³ – 2e² + e – 1) + (–3e³ + 4e² – 2e + 3)

Add directly:
= 5e³ – 2e² + e – 1 – 3e³ + 4e² – 2e + 3

Group like terms:
e³: 5e³ – 3e³ = 2e³
e²: –2e² + 4e² = 2e²
e: e – 2e = –e
constants: –1 + 3 = 2

Final answer for #7: 2e³ + 2e² – e + 2

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Problem 8:
(6f² – 4f + 9) – (–2f² + f – 3)

Change signs:
= 6f² – 4f + 9 + 2f² – f + 3

Group like terms:
f²: 6f² + 2f² = 8f²
f: –4f – f = –5f
constants: 9 + 3 = 12

Final answer for #8: 8f² – 5f + 12

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Problem 9:
(–g³ + 2g² – g + 4) + (3g³ – g² + 2g – 5)

Add directly:
= –g³ + 2g² – g + 4 + 3g³ – g² + 2g – 5

Group like terms:
g³: –g³ + 3g³ = 2g³
g²: 2g² – g² = g²
g: –g + 2g = g
constants: 4 – 5 = –1

Final answer for #9: 2g³ + g² + g – 1

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Problem 10:
(8h⁴ – 3h² + h – 2) – (5h⁴ + 2h² – 4h + 1)

Change signs:
= 8h⁴ – 3h² + h – 2 – 5h⁴ – 2h² + 4h – 1

Group like terms:
h⁴: 8h⁴ – 5h⁴ = 3h⁴
h²: –3h² – 2h² = –5h²
h: h + 4h = 5h
constants: –2 – 1 = –3

Final answer for #10: 3h⁴ – 5h² + 5h – 3

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Problem 11:
(9j² – 4j + 7) + (–j² + 3j – 2)

Add directly:
= 9j² – 4j + 7 – j² + 3j – 2

Group like terms:
j²: 9j² – j² = 8j²
j: –4j + 3j = –j
constants: 7 – 2 = 5

Final answer for #11: 8j² – j + 5

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Problem 12:
(10k³ – 5k² + k – 3) – (4k³ + 2k² – 3k + 1)

Change signs:
= 10k³ – 5k² + k – 3 – 4k³ – 2k² + 3k – 1

Group like terms:
k³: 10k³ – 4k³ = 6k³
k²: –5k² – 2k² = –7k²
k: k + 3k = 4k
constants: –3 – 1 = –4

Final answer for #12: 6k³ – 7k² + 4k – 4

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Final Answer:
1. –a² – 3a – 1
2. 2b³ – 5b² + 8b – 11
3. –p² – 10p + 10
4. –¼x² + ½x – ¾
5. 8c³ – 7c + 10
6. 2d⁴ + 2d² – 2
7. 2e³ + 2e² – e + 2
8. 8f² – 5f + 12
9. 2g³ + g² + g – 1
10. 3h⁴ – 5h² + 5h – 3
11. 8j² – j + 5
12. 6k³ – 7k² + 4k – 4