Adding polynomials is combining like terms, the same skill as before, on longer expressions. Subtracting needs one extra piece of care. The standard form from the last lesson keeps everything lined up.
Why it works
Like terms have the same variable part, so they count the same kind of thing: $3x^2 + 5x^2 = 8x^2$ just as $3$ apples and $5$ apples are $8$ apples. Subtraction is adding the opposite, and the opposite of a sum is the sum of the opposites: $-(a + b - c) = -a - b + c$. So subtracting a polynomial changes the sign of every one of its terms, not only the first.
| $x^2$ | $x$ | constant | |
|---|---|---|---|
| $5x^2 - 3x + 2$ | $5$ | $-3$ | $2$ |
| $-(2x^2 + 4x - 7)$ | $-2$ | $-4$ | $+7$ |
| difference | $3$ | $-7$ | $9$ |
Key idea
To add, combine like terms. To subtract, change the sign of every term in the polynomial being subtracted, then add.
Worked examples
Worked example: Addition
Simplify $(4x^2 - 3x + 1) + (-x^2 + 5x - 6)$.
- Group like terms: $(4x^2 - x^2) + (-3x + 5x) + (1 - 6)$.
- Combine: $3x^2 + 2x - 5$.
Answer: $3x^2 + 2x - 5$
Worked example: Subtraction
Simplify $(5x^2 - 3x + 2) - (2x^2 + 4x - 7)$.
- Distribute the minus sign: $5x^2 - 3x + 2 - 2x^2 - 4x + 7$.
- Combine like terms: $3x^2 - 7x + 9$.
- Check with $x = 1$: $(4) - (-1) = 5$ and $3 - 7 + 9 = 5$.
Answer: $3x^2 - 7x + 9$
Common mistake
Changing the sign of only the first term of the polynomial being subtracted, giving $5x^2 - 3x + 2 - 2x^2 + 4x - 7$. Every term inside the parentheses is subtracted. A quick check is to substitute a small number like $x = 1$ into the original and into your answer; they must agree.