Simplifying Polynomial Expressions worksheet with 15 practice problems.

Here are the step-by-step solutions for simplifying each polynomial expression. To solve these, we combine "like terms." Like terms are terms that have the exact same variable raised to the exact same power (for example, $x^2$ and $3x^2$ are like terms, but $x^2$ and $x^3$ are not).

1) $(2x^4 + 5x^3) - (12x + 2x^4)$
* Distribute the negative sign: $2x^4 + 5x^3 - 12x - 2x^4$
* Combine $x^4$ terms: $2x^4 - 2x^4 = 0$
* Remaining terms: $5x^3 - 12x$
* Answer: $5x^3 - 12x$

2) $(2x^4 + 2x^3) - (7x^4 + 6x^3)$
* Distribute the negative sign: $2x^4 + 2x^3 - 7x^4 - 6x^3$
* Combine $x^4$ terms: $2x^4 - 7x^4 = -5x^4$
* Combine $x^3$ terms: $2x^3 - 6x^3 = -4x^3$
* Answer: $-5x^4 - 4x^3$

3) $(12x^4 + 4x^3) - (2x^4 - 6x^4)$
* *Note: The term inside the second parenthesis is likely a typo in the original worksheet ($2x^4 - 6x^4$), but we will simplify it as written.*
* Simplify inside the second parenthesis first: $(2x^4 - 6x^4) = -4x^4$
* Rewrite expression: $(12x^4 + 4x^3) - (-4x^4)$
* Distribute negative: $12x^4 + 4x^3 + 4x^4$
* Combine $x^4$ terms: $12x^4 + 4x^4 = 16x^4$
* Answer: $16x^4 + 4x^3$

4) $14x - 3x^2 - 2(6x^2 + 6x^3)$
* Distribute the $-2$: $14x - 3x^2 - 12x^2 - 12x^3$
* Combine $x^2$ terms: $-3x^2 - 12x^2 = -15x^2$
* Order by highest power: $-12x^3 - 15x^2 + 14x$
* Answer: $-12x^3 - 15x^2 + 14x$

5) $(5x^3 - 3) + 5(2x^2 - 3x^3)$
* Distribute the $5$: $5x^3 - 3 + 10x^2 - 15x^3$
* Combine $x^3$ terms: $5x^3 - 15x^3 = -10x^3$
* Order by highest power: $-10x^3 + 10x^2 - 3$
* Answer: $-10x^3 + 10x^2 - 3$

6) $(4x^2 - 2x) - 2(4x^2 - 2x^3)$
* Distribute the $-2$: $4x^2 - 2x - 8x^2 + 4x^3$
* Combine $x^2$ terms: $4x^2 - 8x^2 = -4x^2$
* Order by highest power: $4x^3 - 4x^2 - 2x$
* Answer: $4x^3 - 4x^2 - 2x$

7) $2(4x - 3x^3) - 3(3x^2 + 4x^3)$
* Distribute the $2$: $8x - 6x^3$
* Distribute the $-3$: $-9x^2 - 12x^3$
* Combine all: $8x - 6x^3 - 9x^2 - 12x^3$
* Combine $x^3$ terms: $-6x^3 - 12x^3 = -18x^3$
* Order by highest power: $-18x^3 - 9x^2 + 8x$
* Answer: $-18x^3 - 9x^2 + 8x$

8) $(2x^4 - 2x) - (2x^4 + 5x^3)$
* Distribute the negative sign: $2x^4 - 2x - 2x^4 - 5x^3$
* Combine $x^4$ terms: $2x^4 - 2x^4 = 0$
* Order by highest power: $-5x^3 - 2x$
* Answer: $-5x^3 - 2x$

9) $2x^3 - (4x^2 + 2x) + x^2$
* Distribute the negative sign: $2x^3 - 4x^2 - 2x + x^2$
* Combine $x^2$ terms: $-4x^2 + x^2 = -3x^2$
* Order by highest power: $2x^3 - 3x^2 - 2x$
* Answer: $2x^3 - 3x^2 - 2x$

10) $x^4 - 2(x^3 + x) + 3x$
* Distribute the $-2$: $x^4 - 2x^3 - 2x + 3x$
* Combine $x$ terms: $-2x + 3x = x$
* Order by highest power: $x^4 - 2x^3 + x$
* Answer: $x^4 - 2x^3 + x$

11) $(2x^4 - x^3) - (4x^4 - x^3)$
* Distribute the negative sign: $2x^4 - x^3 - 4x^4 + x^3$
* Combine $x^4$ terms: $2x^4 - 4x^4 = -2x^4$
* Combine $x^3$ terms: $-x^3 + x^3 = 0$
* Answer: $-2x^4$

12) $4x^2 - 5x^4 + 15x^4 - 12x^2$
* Combine $x^4$ terms: $-5x^4 + 15x^4 = 10x^4$
* Combine $x^2$ terms: $4x^2 - 12x^2 = -8x^2$
* Order by highest power: $10x^4 - 8x^2$
* Answer: $10x^4 - 8x^2$

13) $2x^4 - 5x^4 + 14x^4 - 13x^4$
* All terms are like terms ($x^4$).
* Calculate coefficients: $2 - 5 + 14 - 13$
* $2 - 5 = -3$
* $-3 + 14 = 11$
* $11 - 13 = -2$
* Answer: $-2x^4$

14) $2x^3 + 5x^5 - 7x^3 + 12x$
* Combine $x^3$ terms: $2x^3 - 7x^3 = -5x^3$
* Other terms stay the same.
* Order by highest power: $5x^5 - 5x^3 + 12x$
* Answer: $5x^5 - 5x^3 + 12x$

15) $2x^4 + 5x^4 + 8x^4 - 8x^4$
* All terms are like terms ($x^4$).
* Calculate coefficients: $2 + 5 + 8 - 8$
* $8 - 8 = 0$, so we are left with $2 + 5 = 7$
* Answer: $7x^4$

16) $5x^4 + 15x - x^2 - 2x^4$
* Combine $x^4$ terms: $5x^4 - 2x^4 = 3x^4$
* Other terms stay the same.
* Order by highest power: $3x^4 - x^2 + 15x$
* Answer: $3x^4 - x^2 + 15x$

Final Answer:
1) $5x^3 - 12x$
2) $-5x^4 - 4x^3$
3) $16x^4 + 4x^3$
4) $-12x^3 - 15x^2 + 14x$
5) $-10x^3 + 10x^2 - 3$
6) $4x^3 - 4x^2 - 2x$
7) $-18x^3 - 9x^2 + 8x$
8) $-5x^3 - 2x$
9) $2x^3 - 3x^2 - 2x$
10) $x^4 - 2x^3 + x$
11) $-2x^4$
12) $10x^4 - 8x^2$
13) $-2x^4$
14) $5x^5 - 5x^3 + 12x$
15) $7x^4$
16) $3x^4 - x^2 + 15x$