Polynomials • Topic 2 of 4

Polynomial Operations

Adding and subtracting polynomials means combining like terms — terms with the same variable and exponent. To add, drop the parentheses and combine; to subtract, distribute the minus sign across the second polynomial first, then combine. For (3x² + 2x) − (x² − 4x), the subtraction gives 3x² + 2x − x² + 4x = 2x² + 6x. Line up like terms by degree to avoid missing any. These operations are the basis of simplifying expressions before solving, and the SAT often tests the sign handling when subtracting a polynomial in parentheses.

✅ Solved examples

1. Add (2x² + 3x) + (x² + 5x).
Combine like terms: 3x² + 8x.
2. Subtract (5x² + 2x) − (2x² + x).
Distribute the minus: 5x² + 2x − 2x² − x = 3x² + x.
3. Simplify (x² − 4) + (3x² + 4).
Combine: 4x² + 0 = 4x².
4. Subtract (4x − 1) − (x − 6).
Distribute the minus: 4x − 1 − x + 6 = 3x + 5.

✏️ Practice — try these, take hints as needed

1. Add (3x² + x) + (2x² + 4x).
Combine like terms.
3x² + 2x² and x + 4x.
—
5x² + 5x.
2. Subtract (6x² + 3x) − (2x² + x).
Distribute the minus.
6x² − 2x² and 3x − x.
—
4x² + 2x.
3. Subtract (5x − 2) − (3x − 7).
Distribute the minus sign.
5x − 3x and −2 + 7.
—
2x + 5.
4. Add (x² + 2x + 1) + (x² − 2x + 3).
Combine like terms by degree.
x² + x², 2x − 2x, 1 + 3.
—
2x² + 4.
5. Subtract (7x² + x) − (7x² − x).
Distribute the minus.
7x² − 7x² and x + x.
—
2x.

📝 Topic test — 8 questions

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