Worksheet: Polynomials - Multiply and Simplify Algebraic - Free Printable

Explanation:
We are asked to simplify each polynomial by combining like terms. Like terms are terms that have the same variable raised to the same power (e.g., $5x^2$ and $-3x^2$ are like terms; $4x$ and $7x^2$ are not).

Let’s go through each problem one by one, carefully grouping and adding coefficients for each power of $x$: $x^3$, $x^2$, $x$, and constants.

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1) $(16x^3 + 7x^2 - 7x^3) + (23x^2 - 23x^3 + 14x^2)$
Group like terms:
- $x^3$: $16x^3 - 7x^3 - 23x^3 = (16 - 7 - 23)x^3 = -14x^3$
- $x^2$: $7x^2 + 23x^2 + 14x^2 = (7 + 23 + 14)x^2 = 44x^2$
No $x$ or constant terms.
→ $-14x^3 + 44x^2$

2) $3x + 3x^2 + x^3 + 3x - 3$
Group:
- $x^3$: $x^3$
- $x^2$: $3x^2$
- $x$: $3x + 3x = 6x$
- constant: $-3$
→ $x^3 + 3x^2 + 6x - 3$

3) $11x^3 + 7x^2 - 5x^3 + 16x^2 - 18x^3 + 12x^2$
- $x^3$: $11 - 5 - 18 = -12$ → $-12x^3$
- $x^2$: $7 + 16 + 12 = 35$ → $35x^2$
→ $-12x^3 + 35x^2$

4) $(10x^3 + 5x^2 - 5x^3) + (15x^2 - 15x^3 + 10x^2)$
First simplify inside parentheses:
- First group: $10x^3 - 5x^3 + 5x^2 = 5x^3 + 5x^2$
- Second: $15x^2 + 10x^2 - 15x^3 = 25x^2 - 15x^3$
Now combine:
- $x^3$: $5x^3 - 15x^3 = -10x^3$
- $x^2$: $5x^2 + 25x^2 = 30x^2$
→ $-10x^3 + 30x^2$

5) $8x^3 + 5x^2 - 2x^3 + 10x^2 - 13x^3 + 7x^2$
- $x^3$: $8 - 2 - 13 = -7$ → $-7x^3$
- $x^2$: $5 + 10 + 7 = 22$ → $22x^2$
→ $-7x^3 + 22x^2$

6) $4x + 7x^2 - x^3 + 5x$
- $x^3$: $-x^3$
- $x^2$: $7x^2$
- $x$: $4x + 5x = 9x$
→ $-x^3 + 7x^2 + 9x$

7) $(13x^3 + 4x^2 - 2x^3) + (15x^2 - 17x^3 + 6x^2)$
First group:
- First: $13x^3 - 2x^3 + 4x^2 = 11x^3 + 4x^2$
- Second: $15x^2 + 6x^2 - 17x^3 = 21x^2 - 17x^3$
Combine:
- $x^3$: $11 - 17 = -6$ → $-6x^3$
- $x^2$: $4 + 21 = 25$ → $25x^2$
→ $-6x^3 + 25x^2$

8) $20x + 3x^2 + x^3 + 4x - 4$
- $x^3$: $x^3$
- $x^2$: $3x^2$
- $x$: $20x + 4x = 24x$
- const: $-4$
→ $x^3 + 3x^2 + 24x - 4$

9) $9x^3 + 4x^2 - 5x^3 + 14x^2 - 13x^3 + 9x^2$
- $x^3$: $9 - 5 - 13 = -9$ → $-9x^3$
- $x^2$: $4 + 14 + 9 = 27$ → $27x^2$
→ $-9x^3 + 27x^2$

10) $(7x^3 + 6x^2 - 5x^3) + (12x^2 - 13x^3 + 11x^2)$
First group:
- First: $7x^3 - 5x^3 + 6x^2 = 2x^3 + 6x^2$
- Second: $12x^2 + 11x^2 - 13x^3 = 23x^2 - 13x^3$
Combine:
- $x^3$: $2 - 13 = -11$ → $-11x^3$
- $x^2$: $6 + 23 = 29$ → $29x^2$
→ $-11x^3 + 29x^2$

11) $10x^3 + 7x^2 - 2x^3 + 12x^2 - 17x^3 + 9x^2$
- $x^3$: $10 - 2 - 17 = -9$ → $-9x^3$
- $x^2$: $7 + 12 + 9 = 28$ → $28x^2$
→ $-9x^3 + 28x^2$

12) $(6x^3 + 7x^2 - 7x^3) + (13x^2 - 13x^3 + 14x^2)$
First:
- First: $6x^3 - 7x^3 + 7x^2 = -x^3 + 7x^2$
- Second: $13x^2 + 14x^2 - 13x^3 = 27x^2 - 13x^3$
Combine:
- $x^3$: $-1 -13 = -14$ → $-14x^3$
- $x^2$: $7 + 27 = 34$ → $34x^2$
→ $-14x^3 + 34x^2$

13) $10x + 6x^2 - x^3 + 7x$
- $x^3$: $-x^3$
- $x^2$: $6x^2$
- $x$: $10x + 7x = 17x$
→ $-x^3 + 6x^2 + 17x$

14) $19x^3 + 5x^2 - 4x^3 + 23x^2 - 24x^3 + 9x^2$
- $x^3$: $19 - 4 - 24 = -9$ → $-9x^3$
- $x^2$: $5 + 23 + 9 = 37$ → $37x^2$
→ $-9x^3 + 37x^2$

15) $2x + 5x^2 + x^3 + 3x - 3$
- $x^3$: $x^3$
- $x^2$: $5x^2$
- $x$: $2x + 3x = 5x$
- const: $-3$
→ $x^3 + 5x^2 + 5x - 3$

16) $6x + 4x^2 + x^3 + 7x - 7$
- $x^3$: $x^3$
- $x^2$: $4x^2$
- $x$: $6x + 7x = 13x$
- const: $-7$
→ $x^3 + 4x^2 + 13x - 7$

17) $(11x^3 + 6x^2 - 4x^3) + (15x^2 - 17x^3 + 10x^2)$
First:
- First: $11x^3 - 4x^3 + 6x^2 = 7x^3 + 6x^2$
- Second: $15x^2 + 10x^2 - 17x^3 = 25x^2 - 17x^3$
Combine:
- $x^3$: $7 - 17 = -10$ → $-10x^3$
- $x^2$: $6 + 25 = 31$ → $31x^2$
→ $-10x^3 + 31x^2$

18) $(5x^3 + 5x^2 - 2x^3) + (7x^2 - 10x^3 + 7x^2)$
First:
- First: $5x^3 - 2x^3 + 5x^2 = 3x^3 + 5x^2$
- Second: $7x^2 + 7x^2 - 10x^3 = 14x^2 - 10x^3$
Combine:
- $x^3$: $3 - 10 = -7$ → $-7x^3$
- $x^2$: $5 + 14 = 19$ → $19x^2$
→ $-7x^3 + 19x^2$

19) $(14x^3 + 7x^2 - 4x^3) + (18x^2 - 21x^3 + 11x^2)$
First:
- First: $14x^3 - 4x^3 + 7x^2 = 10x^3 + 7x^2$
- Second: $18x^2 + 11x^2 - 21x^3 = 29x^2 - 21x^3$
Combine:
- $x^3$: $10 - 21 = -11$ → $-11x^3$
- $x^2$: $7 + 29 = 36$ → $36x^2$
→ $-11x^3 + 36x^2$

20) $15x + 3x^2 - x^3 + 4x$
- $x^3$: $-x^3$
- $x^2$: $3x^2$
- $x$: $15x + 4x = 19x$
→ $-x^3 + 3x^2 + 19x$

21) $7x + 7x^2 - x^3 + 7x$
- $x^3$: $-x^3$
- $x^2$: $7x^2$
- $x$: $7x + 7x = 14x$
→ $-x^3 + 7x^2 + 14x$

22) $18x^3 + 3x^2 - 4x^3 + 22x^2 - 21x^3 + 7x^2$
- $x^3$: $18 - 4 - 21 = -7$ → $-7x^3$
- $x^2$: $3 + 22 + 7 = 32$ → $32x^2$
→ $-7x^3 + 32x^2$

23) $9x + 3x^2 - x^3 + 7x$
- $x^3$: $-x^3$
- $x^2$: $3x^2$
- $x$: $9x + 7x = 16x$
→ $-x^3 + 3x^2 + 16x$

24) $21x + 5x^2 - x^3 + 3x$
- $x^3$: $-x^3$
- $x^2$: $5x^2$
- $x$: $21x + 3x = 24x$
→ $-x^3 + 5x^2 + 24x$

25) $18x + 3x^2 + x^3 + 3x - 3$
- $x^3$: $x^3$
- $x^2$: $3x^2$
- $x$: $18x + 3x = 21x$
- const: $-3$
→ $x^3 + 3x^2 + 21x - 3$

26) $17x + 2x^2 - x^3 + 5x$
- $x^3$: $-x^3$
- $x^2$: $2x^2$
- $x$: $17x + 5x = 22x$
→ $-x^3 + 2x^2 + 22x$

27) $12x + 4x^2 + x^3 + 3x - 3$
- $x^3$: $x^3$
- $x^2$: $4x^2$
- $x$: $12x + 3x = 15x$
- const: $-3$
→ $x^3 + 4x^2 + 15x - 3$

28) $8x + 2x^2 - x^3 + 5x$
- $x^3$: $-x^3$
- $x^2$: $2x^2$
- $x$: $8x + 5x = 13x$
→ $-x^3 + 2x^2 + 13x$

29) $20x^3 + 7x^2 - 3x^3 + 23x^2 - 27x^3 + 10x^2$
- $x^3$: $20 - 3 - 27 = -10$ → $-10x^3$
- $x^2$: $7 + 23 + 10 = 40$ → $40x^2$
→ $-10x^3 + 40x^2$

30) $5x + 4x^2 + x^3 + 4x - 4$
- $x^3$: $x^3$
- $x^2$: $4x^2$
- $x$: $5x + 4x = 9x$
- const: $-4$
→ $x^3 + 4x^2 + 9x - 4$

All simplified.

Final Answer:
1) $-14x^3 + 44x^2$
2) $x^3 + 3x^2 + 6x - 3$
3) $-12x^3 + 35x^2$
4) $-10x^3 + 30x^2$
5) $-7x^3 + 22x^2$
6) $-x^3 + 7x^2 + 9x$
7) $-6x^3 + 25x^2$
8) $x^3 + 3x^2 + 24x - 4$
9) $-9x^3 + 27x^2$
10) $-11x^3 + 29x^2$
11) $-9x^3 + 28x^2$
12) $-14x^3 + 34x^2$
13) $-x^3 + 6x^2 + 17x$
14) $-9x^3 + 37x^2$
15) $x^3 + 5x^2 + 5x - 3$
16) $x^3 + 4x^2 + 13x - 7$
17) $-10x^3 + 31x^2$
18) $-7x^3 + 19x^2$
19) $-11x^3 + 36x^2$
20) $-x^3 + 3x^2 + 19x$
21) $-x^3 + 7x^2 + 14x$
22) $-7x^3 + 32x^2$
23) $-x^3 + 3x^2 + 16x$
24) $-x^3 + 5x^2 + 24x$
25) $x^3 + 3x^2 + 21x - 3$
26) $-x^3 + 2x^2 + 22x$
27) $x^3 + 4x^2 + 15x - 3$
28) $-x^3 + 2x^2 + 13x$
29) $-10x^3 + 40x^2$
30) $x^3 + 4x^2 + 9x - 4$