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Stage 2 · Algebra Part 1

Adding & Subtracting Polynomials

Adding polynomials is straightforward. Subtracting is where most students lose points — because subtraction is really just distributing a negative sign. Get that right and the rest is combining like terms.

How do you want to start?

📖 Learn the Skill Lesson + examples ✏️ Foundations Practice Practice questions 🎯 GED Level Practice GED level questions

📖 Quick Review

A polynomial is an expression made of terms connected by + or − signs. You already know how to combine like terms. That's all adding and subtracting polynomials is — once you handle the parentheses correctly.

Binomial
Two terms — example: x + 3 or 3x − 7
Trinomial
Three terms — example: x² + 3x + 2

Adding Polynomials

When adding, the parentheses come off and nothing changes. Just remove them and combine like terms.

Worked Example — Binomial + Binomial

Simplify: (−3x + 5) + (2x − 8)

(−3x + 5) + (2x − 8)
→ Adding — parentheses come off, nothing changes
−3x + 5 + 2x − 8
→ Combine x terms: −3x + 2x = −x
→ Combine constants: 5 + (−8) = −3
= −x − 3
Worked Example — Trinomial + Trinomial

Simplify: (−x² + 3x − 4) + (3x² − 5x + 1)

(−x² + 3x − 4) + (3x² − 5x + 1)
→ Remove parentheses — addition changes nothing
−x² + 3x − 4 + 3x² − 5x + 1
→ Combine x² terms: −x² + 3x² = 2x²
→ Combine x terms: 3x + (−5x) = −2x
→ Combine constants: −4 + 1 = −3
= 2x² − 2x − 3

Subtracting Polynomials — The Hidden Skill

Subtraction is where most students lose points. When you subtract a polynomial, you are distributing a negative sign to every term inside the second set of parentheses. Every sign flips.

The rule: (A) − (B) means (A) + (−B). Rewrite subtraction as adding the opposite, then distribute the negative to every term inside.
Worked Example — Binomial − Binomial

Simplify: (4x − 7) − (−3x + 5)

(4x − 7) − (−3x + 5)
→ Rewrite as adding the opposite
→ Distribute the negative: −(−3x + 5) = +3x − 5
→ Subtracting a negative x makes it positive
4x − 7 + 3x − 5
→ Combine x terms: 4x + 3x = 7x
→ Combine constants: −7 − 5 = −12
= 7x − 12
Worked Example — Trinomial − Trinomial

Simplify: (−2x² + 3x − 1) − (x² − 2x + 5)

(−2x² + 3x − 1) − (x² − 2x + 5)
→ Distribute the negative to all three terms in the second group
→ −(x² − 2x + 5) = −x² + 2x − 5
→ The −2x becomes +2x because −(−2x) = +2x
−2x² + 3x − 1 − x² + 2x − 5
→ Combine x² terms: −2x² − x² = −3x²
→ Combine x terms: 3x + 2x = 5x
→ Combine constants: −1 − 5 = −6
= −3x² + 5x − 6

Step Strategy

1
Rewrite subtraction as adding the opposite — change the − sign to + and flip every sign inside the second group
2
Remove the parentheses — once you've distributed the negative, the grouping symbols come off
3
Combine like terms — x² terms together, x terms together, constants together
4
Write in standard form — highest exponent first
Common mistakes:
• Only flipping the first sign in the second group — every term must flip
• Forgetting that −(−2) = +2
• Combining unlike terms — x² and x are not the same
• Losing track of negative signs when combining
🎯 Test Strategy — Use the Answer Choices as Checkpoints

On the GED you always have four answer choices. Use them to check your work as you go — not just at the end.

1.Before you start, look at the answer choices. What is the highest exponent? Every answer that doesn't start with that term is already wrong. Cross them out mentally.
2.Work out the highest term first. Once you have it, eliminate any answer choice that doesn't match. You may already be down to two choices.
3.Work out the x term next. Compare to your remaining choices and eliminate again.
4.The constant at the end is usually the tiebreaker. If you've been careful with signs, it will match exactly one answer.

You're not guessing — you're solving in order and letting the answer choices confirm each step. If your x² term doesn't match any answer, you know to stop and recheck before going further.

Reference
GED Formula Sheet
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📋 Open Formula Sheet

✏️ Practice Questions

Bank 1 — Adding Polynomials
Questions are random — answer as many as you like
You've answered 5 questions. Keep going or check your score.
Bank 2 — Subtracting Polynomials
Questions are random — answer as many as you like
You've answered 5 questions. Keep going or check your score.
Up Next in Algebra Part 1
Multiply Polynomials
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