Adding Polynomials Worksheets - Free Printable

Here are the solutions for adding the polynomials. To solve these, we group "like terms" together. Like terms are terms that have the exact same variable and exponent (for example, $x^2$ and $x^2$ go together, but $x^2$ and $x$ do not).

1) $(3x^2 + 9x + 2) + (4x + 3 - 5x^2)$
* Group $x^2$ terms: $3x^2 - 5x^2 = -2x^2$
* Group $x$ terms: $9x + 4x = 13x$
* Group numbers: $2 + 3 = 5$
* Result: $-2x^2 + 13x + 5$

2) $(4a^3 + a^2 - 8a^2 + a + 6) + (-4a^3 + a^2 - 8a + 6)$
* Group $a^3$ terms: $4a^3 - 4a^3 = 0$
* Group $a^2$ terms: $a^2 - 8a^2 + a^2 = -6a^2$
* Group $a$ terms: $a - 8a = -7a$
* Group numbers: $6 + 6 = 12$
* Result: $-6a^2 - 7a + 12$

3) $(2m^2 + 5n^2 + 8mn) + (3n^2 + 10m^2 + 4mn + 2n^2)$
* Group $m^2$ terms: $2m^2 + 10m^2 = 12m^2$
* Group $n^2$ terms: $5n^2 + 3n^2 + 2n^2 = 10n^2$
* Group $mn$ terms: $8mn + 4mn = 12mn$
* Result: $12m^2 + 10n^2 + 12mn$

4) $(7c^2 - 2bc + 4b^2 + c^2) + (-c^2 - 2b^2 + 5bc)$
* Group $c^2$ terms: $7c^2 + c^2 - c^2 = 7c^2$
* Group $bc$ terms: $-2bc + 5bc = 3bc$
* Group $b^2$ terms: $4b^2 - 2b^2 = 2b^2$
* Result: $7c^2 + 3bc + 2b^2$

5) $(8p^2 - 6p^2 + 4p - 4 + 6p^2) + (8p^2 - 6p - 6p^2)$
* Group $p^2$ terms: $8p^2 - 6p^2 + 6p^2 + 8p^2 - 6p^2 = 10p^2$
* Group $p$ terms: $4p - 6p = -2p$
* Group numbers: $-4$
* Result: $10p^2 - 2p - 4$

6) $(-2xy^2 + 6y + 14y^2 + 4y^2) + (2y^2 + y^2 - 5y)$
* Group $xy^2$ terms: $-2xy^2$ (no other like terms)
* Group $y^2$ terms: $14y^2 + 4y^2 + 2y^2 + y^2 = 21y^2$
* Group $y$ terms: $6y - 5y = y$
* Result: $-2xy^2 + 21y^2 + y$

7) $(-r^2 + 5rs + 2s^2 + 6) + (-5 + r^2 + 2s^2 - 5rs)$
* Group $r^2$ terms: $-r^2 + r^2 = 0$
* Group $rs$ terms: $5rs - 5rs = 0$
* Group $s^2$ terms: $2s^2 + 2s^2 = 4s^2$
* Group numbers: $6 - 5 = 1$
* Result: $4s^2 + 1$

8) $(6ab^2 + 2a^2 - b + 2d) + (3d - 5ab^2 + 2a - 5d^2)$
* Group $ab^2$ terms: $6ab^2 - 5ab^2 = ab^2$
* Group $a^2$ terms: $2a^2$
* Group $a$ terms: $2a$
* Group $b$ terms: $-b$
* Group $d$ terms: $2d + 3d = 5d$
* Group $d^2$ terms: $-5d^2$
* Result: $ab^2 + 2a^2 + 2a - b + 5d - 5d^2$

Final Answer:
1) $-2x^2 + 13x + 5$
2) $-6a^2 - 7a + 12$
3) $12m^2 + 10n^2 + 12mn$
4) $7c^2 + 3bc + 2b^2$
5) $10p^2 - 2p - 4$
6) $-2xy^2 + 21y^2 + y$
7) $4s^2 + 1$
8) $ab^2 + 2a^2 + 2a - b + 5d - 5d^2$